MV-NIS or Investigator's Choice Chemotherapy in Treating Patients With Ovarian, Fallopian, or Peritoneal Cancer

Inclusion Criteria: * PRE-REGISTRATION INCLUSION CRITERIA: * Ability to understand and the willingness to sign a written informed consent document * The effects of the candidate chemoprevention agents on the developing human fetus remain incompletely defined; however, study participants will be women who have gone through a bi-lateral oophorectomy procedure * Willingness to be evaluated for surgical placement of an intraperitoneal port and undergo biopsy if feasible for a research sample * REGISTRATION/RANDOMIZATION INCLUSION CRITERIA: * Recurrent, persistent, or progressive epithelial ovarian, fallopian tube, or primary peritoneal cancer after treatment with bilateral oophorectomy and either cisplatin or carboplatin and either paclitaxel, albumin-bound paclitaxel, or docetaxel; histologic confirmation of the primary tumor is required; eligible histologies include serous, endometrioid, clear cell, mucinous, transitional cell, undifferentiated, or mixed carcinoma * Platinum-resistant or platinum-refractory disease, defined as either 1) less than a complete response to the most recent carboplatin- or cisplatin-containing chemotherapy regimen, 2) serum cancer antigen (CA)-125 \>= 2 x upper limit of normal (ULN) within 180 days of last dose of carboplatin- or cisplatin-containing chemotherapy, confirmed by a second CA-125 (the second CA-125 does not have to be within 180 days of chemotherapy), or 3) CT or positron emission tomography (PET)/CT evidence of cancer recurrence within 180 days of last dose of carboplatin- or cisplatin-containing chemotherapy * Absolute neutrophil count (ANC) \>= 1500/uL (obtained =\< 7 days prior to registration) * Platelet (PLT) \>= 100,000/uL (obtained =\< 7 days prior to registration) * Total bilirubin =\< ULN (obtained =\< 7 days prior to registration) * Aspartate aminotransferase (AST) =\< 2 x ULN (obtained =\< 7 days prior to registration) * Creatinine =\< 1.5 x ULN (obtained =\< 7 days prior to registration) * Hemoglobin (Hgb) \>= 9.0 g/dL (obtained =\< 7 days prior to registration) * Willingness to return to Mayo Clinic Rochester or another participating institution for follow-up; patients who are randomized to Arm B (cytotoxic chemotherapy) may receive chemotherapy at any oncology clinic able to provide the protocol-directed therapy and willing to send laboratory data to the participating institution; however, patients must be willing to return to the participating institution every two months for evaluation; patients who are randomized to Arm A must be willing to receive all treatment and follow-up at a participating institution * Life expectancy \>= 12 weeks * Willingness to provide all biologic specimens as required by the protocol * Measurable disease by Response Evaluation Criteria in Solid Tumors (RECIST) criteria, or evaluable disease by CA-125; (NOTE: CA-125-evaluable disease is defined as serum CA-125 \>= 2 x ULN that is determined by the treating clinician to be due to recurrent ovarian, fallopian tube, or primary peritoneal cancer) * Normal cardiac function, as determined by left ventricular ejection fraction (LVEF) \>= institutional lower limit of normal on echocardiogram or multi-gated acquisition scan (MUGA) =\< 1 month prior to registration * If liposomal doxorubicin hydrochloride (DOXIL) is selected as the investigator's choice chemotherapy: * Lifetime exposure to doxorubicin =\< 240 mg/m\^2 (or equivalent biologic dose if prior exposure to a different anthracycline) * Candidate for surgical placement of an intraperitoneal port, as determined by a gynecologic oncology surgeon * Must have anti-measles immunity as demonstrated by serum immunoglobulin (Ig)G anti-measles antibody levels of \>= 1.1 EU/ml as determined by BioPlex Measles IgG multiplex flow immunoassay Exclusion Criteria: * REGISTRATION/RANDOMIZATION EXCLUSION CRITERIA: * Epithelial tumors of low malignant potential, stromal tumors, and germ cell tumors of the ovary * Evidence of measurable disease (per Response Evaluation Criteria in Solid Tumors version 1.1 \[RECIST 1.1\]) outside of the peritoneal cavity (ex: mediastinal lymphadenopathy, parenchymal liver metastasis, or symptomatic pleural effusion proven or suspected to be due to cancer) * Note: Asymptomatic pleural effusion with or without minimal pleural involvement as long as there is no measurable disease outside the peritoneum/retroperitoneum is allowed * Bulky metastases, defined as any tumor nodule or lymph nodes \> 5 cm in greatest dimension on axial images on pre-treatment CT, PET/CT, or magnetic resonance imaging (MRI) * Note: patients with bulky (\> 5 cm) disease for whom gross total cytoreduction is deemed feasible by a surgeon (with confirmation by a second surgeon after radiologic review) are eligible for participation in the context of cytoreductive surgery * Resistant to all of the following: DOXIL, gemcitabine hydrochloride (GEM), topotecan hydrochloride (TOPA), and weekly paclitaxel (TAXOL); (NOTE: resistance is defined as either 1) less than a complete response to any chemotherapy regimen containing the agent in question \[consider weekly TAXOL as a separate agent from every-three-week TAXOL\], 2) serum CA-125 \>= 2 x ULN within 180 days of last dose of chemotherapy containing the agent in question, confirmed by a second CA-125 \[the second CA-125 does not have to be within 180 days of chemotherapy\], or 3) CT or PET/CT evidence of cancer recurrence/progression within 180 days of last dose of chemotherapy containing the agent in question; \[for example, if a patient previously received carboplatin and GEM, had a complete response, and had initial evidence of relapse \> 180 days after the last dose of GEM, that patient would not be considered resistant to GEM\]) * Eastern Cooperative Oncology Group (ECOG) performance status (PS) of 3 or 4 * History of other malignancy =\< 5 years prior to registration except for non-melanoma skin cancer, carcinoma in situ of the cervix, and ductal carcinoma in situ (DCIS) * Active infection =\< 7 days prior to study entry * Any of the following prior therapies: * Chemotherapy =\< 3 weeks prior to study entry * Immunotherapy =\< 4 weeks prior to study entry * Biologic therapy =\< 4 weeks prior to study entry * Extensive abdominal surgery if it includes enterotomy(ies) =\< 3 weeks prior to study entry; (NOTE: this criterion does not apply to placement of the peritoneal Port-A-Cath or lysis of adhesions at the time of study entry) * Any viral or gene therapy prior to study entry * Failure to recover to =\< grade 1 from acute, reversible effects of prior chemotherapy, excluding alopecia regardless of interval since last treatment; (NOTE: patients with residual peripheral neuropathy are allowed) * New York Heart Association classification III or IV congestive heart failure, known symptomatic coronary artery disease, symptoms of coronary artery disease on systems review, or known cardiac arrhythmias (atrial fibrillation or supraventricular tachycardia \[SVT\]) * Other cardiac or pulmonary disease that, at the investigator's discretion, can impair treatment safety * Central nervous system (CNS) metastases or seizure disorder * Human immunodeficiency virus (HIV)-positive test result, or history of other immunodeficiency * History of organ transplantation * History of chronic hepatitis B or C * Other concurrent chemotherapy, immunotherapy, radiotherapy, or any ancillary therapy considered investigational (utilized for a non-Food and Drug Administration \[FDA\]-approved indication and in the context of a research investigation) * Any concurrent medications which could interfere with the trial * History of tuberculosis or history of purified protein derivative (PPD) positivity * Treatment with oral/systemic corticosteroids, with the exception of topical or inhaled steroids or steroids given for the purpose of adrenal replacement given at physiologic doses * Exposure to household contacts =\< 15 months old or household contact with known immunodeficiency * Allergy to measles vaccine or history of severe reaction to prior measles vaccination * Allergy to iodine; (NOTE: this does not include reactions to intravenous contrast materials) * Any other pathology or condition which the principal investigator may deem to negatively impact treatment safety * On anticoagulation and unable to discontinue temporarily for up to 7 days