Pembrolizumab and BCG Solution in Treating Patients With Recurrent Non-Muscle-Invasive Bladder Cancer

Inclusion Criteria: * Patients must have a histologically documented recurrence of non-muscle-invasive bladder carcinoma (T1HG, T1HG after repeat transurethral resection \[reTUR\]) or BCG refractory; if patient has received BCG they can be Ta, Tis, or T1) * Patients must have persistent high grade disease OR be BCG refractory, defined as either: * Recurrence within 6 months of receiving at least 2 courses of intravesical BCG (at least 5 or 6 inductions and at least 2 or 3 maintenance doses) or * T1 high grade disease at the first evaluation following induction BCG alone (at least 5 of 6 induction doses) * Patients must agree to provide tissue from archival biopsy samples or newly obtained excisional biopsy of a tumor lesion * NOTE: Patients who do not have available specimens from previous biopsy or do not agree to provide this tissue are not eligible; cytological specimens will not be acceptable; availability of tissue must be confirmed at the time of registration, but the actual sample does not have to be received in order to complete registration * Patients must have received one course of induction treatment with BCG (4-6 weekly doses), irrespective of the interval since last treatment; patients are allowed to have received any number of prior chemotherapy instillations * NOTE: Patients may have received prior intravesical interferon * All patients must have imaging (computed tomography \[CT\] scan or magnetic resonance imaging \[MRI\]) documenting normal upper urinary tracts and absence of locally advanced bladder cancer within 60 days prior to study registration * Have a performance status of 0-2 on the Eastern Cooperative Oncology Group (ECOG) Performance Scale * Absolute neutrophil count (ANC) \>= 1,500 /mcL within 14 days prior to registration * Platelets \>= 100,000 / mcL within 14 days prior to registration * Hemoglobin \>= 9 g/dL or \>= 5.6 mmol/L within 14 days prior to registration * Serum creatinine =\< 1.5 X upper limit of normal (ULN) OR measured or calculated (creatinine clearance should be calculated per institutional standard) creatinine clearance (glomerular filtration rate \[GFR\] can also be used in place of creatinine or creatinine clearance \[CrCl\]) \>= 60 mL/min for subject with creatinine levels \> 1.5 X institutional ULN within 14 days prior to registration * Serum total bilirubin =\< 1.5 X ULN OR direct bilirubin =\< ULN for subjects with total bilirubin levels \> 1.5 ULN within 14 days prior to registration * Aspartate aminotransferase (AST) (serum glutamic-oxaloacetic transaminase \[SGOT\]) and alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 2.5 X ULN within 14 days prior to registration * International normalized ratio (INR) or prothrombin time (PT) =\< 1.5 X ULN unless subject is receiving anticoagulant therapy as long as PT or partial thromboplastin time (PTT) is within therapeutic range of intended use of anticoagulants within 14 days prior to registration * Activated PTT (aPTT) =\< 1.5 X ULN unless subject is receiving anticoagulant therapy as long as PT or PTT is within therapeutic range of intended use of anticoagulants within 14 days prior to registration * Females of child-bearing potential (FOCBP) and males must agree to use adequate contraception (e.g. hormonal or barrier method of birth control; abstinence) prior to study entry, for the duration of study participation, and for 120 days following completion of therapy; should a female patient become pregnant or suspect she is pregnant while participating in this study, she should inform her treating physician immediately; * NOTE: A FOCBP is any woman (regardless of sexual orientation, having undergone a tubal ligation, or remaining celibate by choice) who meets the following criteria * Has not undergone a hysterectomy or bilateral oophorectomy * Has had menses at any time in the preceding 12 consecutive months (and therefore has not been naturally postmenopausal for \> 12 months) * FOCBP must have a negative urine or serum pregnancy test within 7 days prior to receiving the first dose of study medication; if the urine test is positive or cannot be confirmed as negative, a serum pregnancy test will be required * Patients must have the ability to understand and the willingness to sign a written informed consent prior to registration on study Exclusion Criteria: * Patients who have had chemotherapy, targeted small molecule therapy, or radiation therapy within 2 weeks prior to study day -14 or who have not recovered (to =\< grade 1 or baseline) from adverse events due to a previously administered agent are not eligible * Note: subjects with =\< grade 2 neuropathy are an exception to this criterion and do qualify for the study * Note: if subject received major surgery within 4 weeks prior to day -14, they must have recovered adequately from the toxicity and/or complications per PI discretion * Patients may not be receiving any other investigational agents within 4 weeks of the first dose of treatment * Patients who have received a prior monoclonal antibody within 4 weeks prior to study day -14 or who have not recovered (to =\< grade 1 or baseline) from adverse events due to agents administered more than 4 weeks earlier are not eligible * Patients who have a diagnosis of immunodeficiency (per PI discretion) or who have received treatment with systemic immunosuppressive medications (including but not limited to prednisone, cyclophosphamide, azathioprine, methotrexate, thalidomide, and anti-TNF agents within 2 weeks prior to study registration are not eligible * NOTE: patients who have received acute, low-dose, systemic immunosuppressant medications (eg, one-time dose of dexamethasone for nausea) may be enrolled in the study; the use of inhaled corticosteroids and mineralocorticoids (eg, fludrocortisone) is allowed * Patients who have a history of allergic reactions attributed to compounds of similar chemical or biologic composition to MK-3475 are not eligible AND/OR patients who have had prior exposure to compounds of similar chemical or biologic composition to MK-3475 are not eligible * Patients who have documentation of an uncontrolled intercurrent illness (as noted in their medical records) including, but not limited to any of the following, are not eligible * Ongoing or active infection requiring systemic treatment * Symptomatic congestive heart failure (New York Heart Association cardiac disease class III or IV) * Unstable angina pectoris * Myocardial infarction within the previous 3 months * Unstable cardiac arrhythmias * Psychiatric illness/social situations that would limit compliance with study requirements * Any other illness or condition that the treating investigator feels would interfere with study compliance or would compromise the patient's safety or study endpoints * Female patients who are pregnant or nursing are not eligible * Patients who have a history of allergic reactions attributed to compounds of similar chemical or biologic composition to BCG are not eligible * Patients who have had an active infection requiring systemic therapy within 1 week prior to day -14 are not eligible UNLESS they are symptom-free and have a negative culture at the time of dosing on day -14 * Patients who received a live, attenuated vaccine within 4 weeks before study registration or are anticipated to require such a live attenuated vaccine are not eligible; NOTE: Influenza vaccination should be given during influenza season only (approximately October to March); patients must not receive live, attenuated influenza vaccine (e.g., FluMist) within 4 weeks prior to study registration or at any time during the study * Patients who are known to be (i.e. documented in medical records) human immunodeficiency virus (HIV) positive are not eligible * Patients with active tuberculosis are not eligible * Patients with known active hepatitis B (chronic or acute; defined as having a positive hepatitis B surface antigen \[HBsAg\] test at screening) or hepatitis C are not eligible * NOTE: patients with past hepatitis B virus (HBV) infection or resolved HBV infection (defined as the presence of hepatitis B core antibody \[HBc Ab\] and absence of HBsAg) are eligible; HBV deoxyribonucleic acid (DNA) must be obtained in these patients 14 days prior to study registration * NOTE: patients positive for hepatitis C virus (HCV) antibody are eligible only if polymerase chain reaction is negative for HCV ribonucleic acid (RNA) * Patients who have a history of severe allergic, anaphylactic, or other hypersensitivity reactions to chimeric or humanized antibodies or fusion proteins are not eligible * Patients with an active autoimmune disease requiring systemic treatment within the past 2 years (i.e. with use of disease modifying agents, corticosteroids or immunosuppressive drugs); replacement therapy (eg. thyroxine, insulin, or physiologic corticosteroid replacement therapy for adrenal or pituitary insufficiency, etc.) is not considered a form of systemic treatment * Patients with a documented history of clinically severe autoimmune disease, or a syndrome that requires systemic steroids or immunosuppressive agents; subjects with vitiligo or resolved childhood asthma/atopy would be an exception to this rule; subjects that require intermittent use of bronchodilators or local steroid injections would not be excluded from the study; subjects with hypothyroidism stable on hormone replacement or Sjogren's syndrome will not be excluded from the study * Patients with history of interstitial lung disease or active, non-infectious pneumonitis are not eligible * NOTE: history of radiation pneumonitis in the radiation field (fibrosis) is permitted * Treatment with systemic immunostimulatory agents (including but not limited to IFNs, IL-2) within 6 weeks or five half-lives of the drug, whichever is shorter, prior to study registration are not eligible * Patients who received prior treatment with an anti-PD-1, anti-PD-L1, anti-PD-L2, anti-CD137, or anti-cytotoxic T-lymphocyte-associated antigen-4 (CTLA-4) antibody (including ipilimumab or any other antibody or drug specifically targeting T-cell co-stimulation or checkpoint pathways) are not eligible * Patients who have a history of prior malignancy are not eligible; please NOTE the following exceptions when patient has undergone potentially curative therapy with no evidence of that disease recurrence for 5 years since initiation of that therapy * Basal cell carcinoma of the skin * Squamous cell carcinoma of the skin * In situ cervical cancer * Patients who have a history of an allogeneic tissue/solid organ transplant are not eligible