Pembrolizumab/Lenvatinib With and Without Healthy Donor FMT (hdFMT) in Relapsed/ Refractory (R/R) Melanoma

Inclusion Criteria: * Patients with cutaneous melanoma or unknown primary melanoma may enroll. Patients with uveal or mucosal or acral-lentiginous melanoma are excluded. * Male participants: • A male participant must agree to use a contraception as detailed in Appendix 3 of this protocol during the treatment period and for at least 120 days after the last dose of study treatment and refrain from donating sperm during this period. * Female participants: * A female participant is eligible to participate if she is not pregnant; not breastfeeding, and at least one of the following conditions applies: * Not a woman of childbearing potential (WOCBP); OR * A WOCBP who agrees to follow the contraceptive guidanceper protocol during the treatment period and for at least 120 days after the last dose of study treatment. * Participants must have progressed on treatment with an anti-PD(L)1 ICI administered either as monotherapy or in combination with other checkpoint inhibitors or other standard/investigational therapies. PD-1 treatment progression is defined by meeting all the following criteria: * Has received at least 2 doses of an approved anti-PD(L)1 ICI administered as a single agent, in combination with chemotherapy, and/or in combination with other investigational therapy. * Participants who progressed on/within 3 months of adjuvant therapy with anti-PD(L)1 ICI will eligible. * Demonstrated disease progression after anti-PD-1/L1 as defined by RECIST v1.1. The initial evidence of PD is to be confirmed by a second assessment no sooner than 4 weeks from the date of the first documented PD. * Progressive disease has been documented within 12 weeks from the last dose of anti-PD-1/anti-PD-L1 mAb. * NOTE: Progressive disease must be determined as above. * NOTE: This determination is made by the treating investigator. Once PD is confirmed, the initial date of PD documentation will be considered the date of PD. * NOTE: Anti-PD(L)1 ICI need not be the most recent line of therapy administered. * Patients with CNS disease are eligible if CNS metastases are treated and deemed stable prior to date of enrollment. * NOTE: All patients will undergo CNS imaging at the time of Screening. Patients with treated brain metastases will need repeat CNS imaging to document stability. * NOTE: Stability is defined based on appearance of treated lesions on a contrast-enhanced CT or MRI brain study performed as part of screening by radiologist, radiation oncologist or neurosurgeon (whichever is most appropriate); absence of new or enlarging brain metastases; and no longer requiring systemic steroids (≤ 10 mg/day prednisone or equivalent) for at least one week prior to enrollment. * NOTE: The contrast-enhanced CT or MRI brain imaging study should be performed no sooner than 2 weeks after most recent surgical and/or radiological intervention. * NOTE: If lesions were discovered during Screening, the patient may be eligible if the lesions are treated and stable based on the above criteria. * NOTE: Patients with leptomeningeal involvement (leptomeningeal enhancement on MRI/CT imaging and/or positive CSF cytology) are excluded regardless of stability. * Prior treatment(s) • NOTE: Prior anti-CTLA-4 ICI is allowed but not required. • NOTE: Given the clear benefit of immunotherapy (over BRAF/MEK inhibitor therapy) in BRAFV600 mutant melanoma, BRAFV600 mutant patients need not have received BRAF/MEK inhibitor therapy prior to enrollment. * Willingness to repeatedly receive FMT administered endoscopically (colonoscopy or sigmoidoscopy) following necessary bowel preparation pre-procedure. * NOTE: Understands infectious risks associated with FMT administration. o Although FMT infusate has been screened for bacteria, viruses, fungi and parasites there is a risk of transmission of known and unknown infectious organisms contained in the donor stool. Post-FMT bacteremia (e.g. E. coli), sepsis and fatal events may rarely occur. * NOTE: Understands non-infectious risks associated with FMT administration. * Possible allergy and/or anaphylaxis to antigens in donor stool. * Theoretical risk of developing disease possibly related to donor gut microbiota including but not limited to: obesity, metabolic syndrome, cardiovascular disease, autoimmune conditions, allergic/atopic disorders, neurologic disorders, psychiatric conditions and malignancy. * NOTE: Understand risks associated with endoscopy (colonoscopy or sigmoidoscopy) including risk of infection transmission, colonic perforation, aspiration pneumonia, and death. * NOTE: Understand that data regarding the long-term safety risk of FMT are lacking. * Presence of measurable disease based on RECIST 1.1. * Patients need to have at least one measurable lesion and a separate lesion for biopsy. Patients with only 1 lesion may be enrolled after discussion with Sponsor-Investigator. * Lesions situated in a previously irradiated area are considered measurable if progression has been demonstrated in such lesions. * Able to provide newly obtained core or excisional biopsy of a tumor lesion not previously irradiated to undergo tumor biopsy (core, punch, incisional or excisional). • Biopsy must meet minimal sampling criteria as defined in the Schedule of Study Procedures (Section 5.10, Footnote N). * Have an Eastern Cooperative Oncology Group (ECOG) performance status of 0 to 1. * Have adequate organ function per protocol. Specimens must be collected within 28 days prior to the start of study intervention. Criteria for patients with hepatitis B and C: * Screening for hepatitis B and C are required. * For hepatitis B positive patients: * Patients who are hepatitis B positive (i.e. HBsAg positive) or have a history of history of hepatitis B (i.e., HBcAb positive, or history of documented hepatitis B infection) are eligible if they have received hepatitis B directed antiviral therapy for at least 4 weeks and have undetectable HBV viral load (HBV DNA) prior to enrollment. * Participants should remain on antiviral therapy throughout study intervention and follow local guidelines for HBV antiviral therapy post completion of study intervention. * For hepatitis C positive patients: Patients who are hepatitis C positive (i.e. HCV antibody reactive) or have a history of history of hepatitis C (i.e. history of documented hepatitis C infection) are eligible if they have received and completed hepatitis C directed antiviral therapy at least 4 weeks and have undetectable HCV viral load (HCV RNA) prior to enrollment Exclusion Criteria: * Diagnosis of non-cutaneous melanoma histologies including mucosal melanoma, ocular/choroidal melanoma, and acral-lentiginous melanoma. * Prior therapies: * Receipt of prior agent(s) targeting the intestinal microbiome including but not limited to: FMT, defined bacterial consortia, single bacterial species and/or microbiota derived peptides. * Prior chemotherapy, targeted therapy, and/or small molecule therapy within 2 weeks (or 4 half lives) prior to study Day 1. * Prior therapy with lenvatinib or other systemic anti-angiogenic therapy. * Prior radiotherapy within 2 weeks of start of study intervention. * Participants must have recovered from all radiation-related toxicities, not require corticosteroids, and not have had radiation pneumonitis. * A 2-week washout is permitted for palliative radiation (≤2 weeks of radiotherapy) to disease including CNS disease. * Contraindications to receiving lenvatinib: • Has had major surgery within 3 weeks prior to first dose of study interventions. * NOTE: Adequate wound healing after major surgery must be assessed clinically, independent of time elapsed for eligibility. * Has preexisting ≥Grade 3 gastrointestinal or non-gastrointestinal fistula. * Has urine protein ≥1 g/24 hours. Note: Participants with proteinuria ≥2+ (≥100 mg/dL) on urine dipstick testing (urinalysis) will undergo 24-hour urine collection for quantitative assessment of proteinuria. * Has a LVEF below the institutional (or local laboratory) normal range, as determined by multigated acquisition (MUGA) or echocardiogram (TTE). * Has radiographic evidence of encasement or invasion of a major blood vessel, or of intratumoral cavitation. * NOTE: The degree of proximity to major blood vessels should be considered because of the potential risk of severe hemorrhage associated with tumor shrinkage/necrosis following lenvatinib therapy. • Prolongation of QTcF interval to \>480 ms. * NOTE: If the QTcF is prolonged to \>480 ms in the presence of a pacemaker, contact the Sponsor to determine eligibility. • Has clinically significant cardiovascular disease within 12 months from first dose of study intervention, including New York Heart Association Class III or IV congestive heart failure, unstable angina, myocardial infarction, cerebral vascular accident, or cardiac arrhythmia associated with hemodynamic instability. * NOTE: Medically controlled arrhythmia would be permitted. * Gastrointestinal malabsorption or any other condition that might affect the absorption of lenvatinib * Active hemoptysis (bright red blood of at least 0.5 teaspoon) within 3 weeks prior to the first dose of study drug. * Presence of an absolute contraindication(s) to FMT administration * Toxic megacolon * Severe dietary allergies (e.g. shellfish, nuts, seafood) * Inflammatory bowel disease * Patients who have not adequately recovered (i.e., ≤Grade 1 or at baseline or ≤Grade 2 endocrinopathy) from adverse events (AEs) due to a previously administered agent * A WOCBP who has a positive urine pregnancy test at Screening (see Appendix 3). If the urine test is positive or cannot be confirmed as negative, a serum pregnancy test will be required. * Has received a live vaccine within 30 days prior to the first dose of study drug. * Examples of live vaccines include, but are not limited to, the following: measles, mumps, rubella, varicella/zoster (chicken pox), yellow fever, rabies, Bacillus Calmette-Guérin (BCG), and typhoid vaccine. * Seasonal influenza vaccines for injection are generally killed virus vaccines and are allowed; however, intranasal influenza vaccines (ie. FluMist®) are live attenuated vaccines - Has a diagnosis of immunodeficiency, immunosuppression and/or is receiving chronic systemic steroid therapy (in dosing exceeding 10 mg daily of prednisone equivalent) or any other form of immunosuppressive therapy within 14 days prior to the first dose of study drug. * Has active autoimmune disease that has required systemic treatment in the past 2 years (i.e. with use of disease modifying agents, corticosteroids or immunosuppressive drugs). o Note: Replacement therapy (e.g., thyroxine, insulin, or physiologic corticosteroid replacement therapy for adrenal or pituitary insufficiency, etc.) is not considered a form of systemic treatment and is allowed. * Concurrent non-hematologic malignancy within 3 years of data of first planned dose of therapy except for tumors with a negligible risk of metastasis and/or death as defined below: * Adequately treated non-invasive malignancies including but not limited to melanoma in situ (MIS), cutaneous squamous cell carcinoma (cSCC), in situ cSCC, basal cell carcinoma (BCC), CIS of cervix, or DCIS/LCIS of breast. * Low-risk early-stage prostate adenocarcinoma (T1-T2a N0 M0 and Gleason score ≤6 and PSA ≤10 ng/mL) for which the management plan is active surveillance, or prostate adenocarcinoma with biochemical-only recurrence with documented PSA doubling time of \> 12 months for which the management plan is active surveillance. * Indolent hematologic malignancies for which the management plan is active surveillance including but not limited to CLL/indolent lymphoma. * NOTE: Patients with high-risk hematologic malignancies (CML, ALL, AML, Hodgkin's or non-Hodgkin's lymphoma) are excluded even if the management plan is active surveillance. * Active (i.e., symptomatic or growing) central nervous system (CNS) metastases. * Note: Participants with previously treated brain metastases may participate provided they are radiologically stable, i.e. without evidence of progression for at least 2 weeks by repeat imaging (note that the repeat imaging should be performed during Screening), clinically stable and without requirement of steroid treatment for at least 14 days prior to first dose of study intervention * Note: Patients with leptomeningeal disease are excluded. * Has severe hypersensitivity (≥Grade 3) to anti-PD(L)1 inhibitor. Has a systemic disease that requires systemic pharmacologic doses of corticosteroids greater than 10 mg daily prednisone (or equivalent). * Note: Participants who are currently receiving steroids at a dose of ≤10 mg daily do not need to discontinue steroids prior to enrollment. * Note: Participants that require topical, ophthalmologic, injected and/or inhalational steroids are not excluded from the study. * Note: Participants with hypothyroidism stable on hormone replacement or Sjogren's syndrome are not excluded from the study. * Note: Participants who require active immunosuppression (greater than steroid dose discussed above) for any reason are excluded from the study. * Has a history of interstitial lung disease or active, non-infectious pneumonitis that required steroids or has current pneumonitis. * Active infections: * Any active infection requiring systemic therapy. * Active TB (Bacillus Tuberculosis). * Active COVID-19 infection and/or exposure to SARS-CoV-2 as defined below: * Positive SARS-CoV-2 result on nasopharyngeal and/or stool specimens (by RT-PCR test) * Active COVID-19 infection (per CDC guidelines) * Exposure to active COVID-19 infected patient (as confirmed using SARS-CoV-2 RT-PCR test or other approved test) as defined per CDC guidelines. * Active human immunodeficiency virus (HIV) infection. o Patients will be evaluated for HIV during screening. * Concurrent active Hepatitis B (defined as HBsAg positive and/or detectable HBV DNA) and Hepatitis C virus (defined as anti-HCV Ab positive and detectable HCV RNA) infection. * Has a history or current evidence of any condition, therapy, or laboratory abnormality that might confound the results of the study, interfere with the participant's participation for the full duration of the study, or is not in the best interest of the participant to participate, in the opinion of the treating investigator. * Has known psychiatric or substance abuse disorders that would interfere with cooperation with the requirements of the trial. * Is pregnant or breastfeeding or expecting to conceive or father children within the projected duration of the study, starting with the Screening visit through 120 days after the last dose of trial treatment. * Has had an allogenic tissue/solid organ transplant.