Niraparib and Copanlisib in Treating Patients With Recurrent Endometrial, Ovarian, Primary Peritoneal, or Fallopian Tube Cancer

Inclusion Criteria: * Any histologically confirmed recurrent endometrial adenocarcinoma (except for carcinosarcoma), recurrent high-grade serous ovarian/primary peritoneal/fallopian tube carcinoma, or deleterious BRCA mutant recurrent ovarian/primary peritoneal/fallopian tube cancer for whom no curative option is available will be eligible. DOSE ESCALATION ONLY: For patients with ovarian cancer in the dose escalation cohort, only patients with recurrent, platinum resistant disease are eligible to enroll on study. Platinum resistance is defined as progression within 6 months from completion of platinum based therapy (the date should be calculated from the last administered dose of platinum therapy). In addition, patients in this cohort with a BRCA mutation must have also progressed after treatment with PARP inhibitor. * Patients may have unlimited prior chemotherapeutic regimens for management of recurrent endometrial or ovarian carcinoma. Patients who have received prior PARP inhibitors ARE allowed to participate. Patients who have received prior PI3K-pathway inhibitors ARE allowed to participate on the dose escalation phase ONLY. Patients may have progressed on prior PARP inhibitor and/or PI3K-pathway inhibitor but they may not have discontinued drug for toxicity. * With the exception of alopecia, peripheral neuropathy, and bone marrow parameters, any unresolved toxicities from prior chemotherapy should be no greater than Common Terminology Criteria for Adverse Events (CTCAE) (version 5.0) grade 1 at the time of starting study treatment. * Patients should have measurable disease as defined by Response Evaluation Criteria in Solid Tumors (RECIST) 1.1. If no measurable disease is present, patients should have assessable disease such as pleural effusion, ascites, with CA125 Gynecological Cancer Intergroup (GCIG) criteria. * Patients must have an Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1. * The effects of niraparib and copanlisib on the developing human fetus are unknown. For this reason, women of childbearing potential must have a negative urine or serum pregnancy test within 7 days prior to taking study treatment. Female patients and their male partners must also agree to use effective contraception when sexually active. This applies since signing of the informed consent form and 6 months (for women of child bearing potential) after the last study drug administration. A woman is considered of childbearing potential (WOCBP), i.e. fertile, following menarche and until becoming post-menopausal unless permanently sterile. Permanent sterilization methods include but are not limited to hysterectomy, bilateral salpingectomy and bilateral oophorectomy. A postmenopausal state is defined as no menses for 12 months without an alternative medical cause. A high follicle stimulating hormone (FSH) level in the postmenopausal range may be used to confirm a post-menopausal state in women not using hormonal contraception or hormonal replacement therapy. * A man is considered fertile after puberty unless permanently sterile by bilateral orchidectomy. The investigator or a designated associate must advise the patient (WOCBP or men who have not undergone bilateral orchidectomy) how to achieve highly effective birth control method. Birth control should be used from the signing of the patient consent form and for 180 days following the last dose of niraparib. Acceptable methods of birth control include: * Two highly effective forms of contraception, defined as contraceptive methods with a failure rate of less than 1% per year when used consistently and correctly. Patients and their sexual partners who've undergone vasectomy or tubal occlusion must also use a male condom with spermicide. * Permanent sterilization, defined as hysterectomy, bilateral salpingectomy, bilateral oophorectomy, or bilateral orchidectomy; postmenopausal, defined as a female patient or sexual partner \> 45 years of age who has not menstruated for at least 12 consecutive months; total sexual abstinence. * It is unknown if niraparib or copanlisib are expressed in human breast milk. For this reason, women must not breast-feed while taking the study medications. * Absolute neutrophil count \>= 1,500/mcL (within 7 days prior to entry/randomization). * Hemoglobin \>= 9 gm/dL (within 7 days prior to entry/randomization). * Platelets \>= 100,000/mcL (within 7 days prior to entry/randomization). * Total bilirubin =\< 1.5 x institutional upper limit of normal (ULN) (within 7 days prior to entry/randomization) (\< 2 x ULN for patients with Gilbert-Meulengracht syndrome or for patients with cholestasis due to compressive adenopathies of the hepatic hilum). * Aspartate aminotransferase (AST)/alanine aminotransferase (ALT) =\< 2.5 x ULN unless the liver is involved with tumor, in that case, ALT/AST must be =\< 5 x ULN (within 7 days prior to entry/randomization). * Lipase =\< 1.5 x ULN (within 7 days prior to entry/randomization). * International normalized ratio (INR) =\< 1.5 x ULN and partial thromboplastin time (PTT) =\< 1.5 x ULN (within 7 days prior to entry/randomization). Prothrombin time (PT) can be used instead of INR if PTT =\< 1.5 x ULN. * Left ventricular ejection fraction (LVEF) \>= 50% (within 7 days prior to entry/randomization). * Glomerular filtration rate (GFR) \>= 50 mL/min/1.73 m\^2 according to the Chronic Kidney Disease Epidemiology Collaboration (CKD-EPI) equation. If not on target, the evaluation may be repeated once after at least 24 hours either according to the CKD-EPI equation or by 24 hour sampling. If the later result is within acceptable range, it may be used to fulfill the inclusion criteria instead. * Patients must be able to swallow and tolerate oral medications and not have gastrointestinal illnesses that would preclude absorption of niraparib (e.g. uncontrolled nausea, vomiting, or diarrhea; malabsorption syndrome; ulcerative disease). * Participants' life expectancy must be at least 3 months. * Patients must be able to understand and willing to sign an informed consent. * Patients must agree to not donate blood during the study or for 90 days after the last dose of study treatment. * For the expansion phase only: For the expansion phase, patients with recurrent platinum sensitive disease must progress after treatment with a platinum doublet, and patients with BRCA mutations must progress after maintenance therapy with PARP inhibitor. * For the expansion phase only: For the expansion phase, patients must have measurable disease accessible for biopsy. * For the expansion phase only: Archival specimens from the time of primary or recurrence diagnosis. Exclusion Criteria: * Patients must not be simultaneously enrolled in an interventional clinical trial. * Patients with endometrial cancer may not have carcinosarcoma. * Patients who have recurrences that are amenable to potentially curative treatment with radiation therapy or surgery. * Patients who have a history of other malignancies except for basal cell or squamous cell skin cancer, in situ cervical cancer, unless they have been disease-free for at least three years. Patients may have dual primaries of ovarian and endometrial cancer, superficial bladder, or localized prostate cancer. * Patients who have had prior chemotherapy, biological therapy, radiation therapy, androgens, thalidomide, immunotherapy, other anticancer agents, and any investigational agents within 28 days of starting study treatment or autologous transplant less than 3 months before start of treatment unless evidence of progression since last treatment (not including palliative radiotherapy at focal sites) or within a time interval less than at least 5 half-lives of the investigational agent, whichever is longer, prior to the first scheduled day of dosing in this study. * Patients who have had major surgery within 3 weeks prior to entry into the study or be recovering from any effects of surgery. Patients may not have had minor surgery within 2 weeks or open biopsy less than 7 days prior to entry into the study. * Patients with known hypersensitivity to niraparib or copanlisib or any of their excipients. Patients may not have history of hypersensitivity to drugs with a similar chemical structure or class to niraparib or copanlisib. * Patients who have experienced intolerable adverse events per treating investigator due to other PARP inhibitors or PI3K-pathway inhibitors. * Patients with known history of cancer involvement of the central nervous system. A scan to confirm absence of brain metastasis is not required. * Congestive heart failure \> New York Heart Association (NYHA) class 2. * Unstable angina (angina symptoms at rest), new-onset angina (begun within the last 3 months), myocardial infarction less than 6 months before start of test drug Uncontrolled hypertension (despite optimal medical management). * Arterial or venous thrombotic or embolic events such as cerebrovascular accident (including transient ischemic attacks), deep vein thrombosis or pulmonary embolism within 3 months before the start of study treatment. * Non-healing wound, ulcer, or bone fracture. * Glycosylated hemoglobin (HbA1c) \> 8.5% at screening. * Known history of human immunodeficiency virus (HIV) infection. All patients must be screened for HIV up to 28 days prior to study drug start using a blood test for HIV according to local regulations. * Hepatitis B (HBV) or hepatitis C (HCV). All patients must be screened for HBV and HCV up to 28 days prior to study drug start using the routine hepatitis virus laboratorial panel. Patients positive for hepatitis B surface antigen (HBsAg) or hepatitis B virus core antibody (HBcAb) will be eligible if they are negative for HBV-deoxyribonucleic acid (DNA); patients positive for anti-HCV antibody will be eligible if they are negative for HCV-ribonucleic acid (RNA). * Patients with seizure disorder requiring medication. * Proteinuria of \>= Common Terminology Criteria for Adverse Events (CTCAE) (version 5.0) grade 3 as assessed by a 24 hour total urine protein (estimated by urine protein : creatinine ratio 3.5 on a random urine sample). * History or concurrent condition of interstitial lung disease of any severity and/or severely impaired lung function (as judged by the investigator). * Concurrent diagnosis of pheochromocytoma. * Unresolved toxicity higher than CTCAE (version 5.0) grade 1 attributed to any prior therapy/procedure, excluding alopecia, peripheral neuropathy, and bone marrow parameters. * Any illness or medical conditions that are unstable or could jeopardize the safety of patients and their compliance in the study. * As judged by the investigator, the patient is unsuitable to participate in the study and the patient is unlikely to comply with study procedures, restrictions, and requirements. * Active, clinically serious infections \> CTCAE (version 5.0) grade 2. * History of, or current autoimmune disease, including autoimmune thyroid disease. * Cytomegalovirus (CMV) infection. Patients who are CMV polymerase chain reaction (PCR) positive at baseline will not be eligible. * Patients with evidence or history of bleeding diathesis. Any hemorrhage or bleeding event \>= CTCAE (version 5.0) grade 3 within 4 weeks prior to the start of the study medication. * Pregnant or breast feeding women. Women of childbearing potential must have a serum pregnancy test performed a maximum of 7 days before start of treatment, and a negative result must be documented before start of treatment. * History of having received an allogenic bone marrow or organ transplant. * Patient has any known history of myelodysplastic syndrome (MDS) or acute myeloid leukemia (AML). * Patients has had radiation therapy encompassing \> 20% of the bone marrow within 2 weeks or any radiation therapy within 1 week prior to day 1 of protocol therapy. * Patient must not have received colony stimulating factors (e.g., granulocyte colony stimulating factor, granulocyte macrophage colony stimulating factor, or recombinant erythropoietin) within 4 weeks prior initiating protocol therapy. * EXCLUDED PRIOR THERAPIES AND MEDICATIONS: - Myeloid growth factors less than 14 days before start of treatment - Blood or platelet transfusion less than 7 days before start of treatment - Ongoing systemic corticosteroid therapy at a daily dose higher than 15 mg prednisone or equivalent. Previous corticosteroid therapy must be stopped or reduced to the allowed dose 7 days before performing the screening computed tomography (CT)/magnetic resonance imaging (MRI) and again prior to the first study drug administration. If a patient is on chronic corticosteroid therapy, corticosteroids should be de-escalated to the maximum allowed dose before the screening. Patients may use topical or inhaled corticosteroids. - Anti-arrhythmic therapy (beta blockers or digoxin are permitted) - Use of strong inhibitors and inducers of CYP3A4 is prohibited from day -14 of cycle 1 until the end of study visit. Copanlisib is primarily metabolized by CYP3A4. Therefore concomitant use of strong inhibitors of CYP3A4(e.g., ketoconazole, itraconazole, clarithromycin, ritonavir, indinavir, nelfinavir and saquinavir), and inducers of CYP3A (e.g. rifampin, phenytoin, carbamazepine, phenobarbital, St. John's wort) are not permitted from day -14 of cycle 1 until the end of study visit. * For expansion phase only: Lack of accessible tumor for biopsy.