Bevacizumab and Anetumab Ravtansine or Paclitaxel in Treating Patients With Refractory Ovarian, Fallopian Tube, or Primary Peritoneal Cancer

Inclusion Criteria: * Patients must have histologically or cytologically confirmed high grade serous or high grade endometrioid ovarian, fallopian tube, primary peritoneal cancer * Patients must have platinum resistant (platinum-free interval \< 6 months) or platinum refractory disease as per Gynecological Cancer Intergroup (GCIC) criteria * Patients must have radiologic evidence of disease progression * Criteria only for the randomized phase 2 part: patients must have measurable disease, defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded for non-nodal lesions and short axis for nodal lesions) as \>= 20 mm with conventional techniques or as \>= 10 mm with spiral computed tomography (CT) scan, magnetic resonance imaging (MRI), or calipers by clinical exam * Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 (Karnofsky \>= 60%) * Leukocytes \>= 3 x 10\^9/L * Absolute neutrophil count \>= 1.5 x 10\^9/L * Platelets \>= 100 x 10\^9/L * Total bilirubin =\< upper limit of normal (ULN) (except in Gilbert's syndrome) * Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamic pyruvic transaminase \[SGPT\]) =\< 2.5 x institutional upper limit of normal (ULN) * Serum creatinine =\< 1.5 x ULN OR creatinine clearance \>= 50 mL/min/1.73 m\^2 for patients with creatinine levels above 1.5 x institutional normal (using the Cockcroft Gault formula) * Urine protein / creatinine ratio (UPCR) =\< 1 * Ongoing prior toxicities related to previous treatments must be recovered to \< grade 2 at the time of registration (with the exception of alopecia, grade 2 peripheral neuropathy or lymphopenia) * Criteria only for the randomized phase 2 part: mesothelin screen positive determined from archival tumor tissue and to be performed centrally. MSLN-positive is defined as \>= 30% of tumor cells with membrane staining intensities \>= 2+. If an archival tumor specimen is unavailable, or unsuitable for mesothelin testing, a pre-treatment fresh tumor biopsy is required * For the run-in-phase 1, patients will not be selected based on the mesothelin expression * Patients with known brain metastases are not excluded from this clinical trial. Patients who received palliative radiation (for brain metastases) are eligible if they have been asymptomatic for at least 4 weeks without use of maintenance steroid therapy, and last received radiation at least 4 weeks prior to proposed start of therapy * Ability to understand and the willingness to sign a written informed consent document. Patients with Impaired Decision Making Capacity (IDMC) can have a Legally Authorized Representative sign on their behalf. Documentation, such as a Power of Attorney, must be presented in order for a substitute decision maker to be allowed Exclusion Criteria: * Patients who have had chemotherapy or radiotherapy within 4 weeks (6 weeks for nitrosoureas or mitomycin C) prior to entering the study * Patients who are receiving any other investigational agents * History of severe allergic reactions attributed to compounds of similar chemical or biologic composition to bevacizumab, anetumab ravtansine or paclitaxel * Patients receiving any medications or substances that are strong inhibitors or inducers of CYP3A4 are ineligible. Because the lists of these agents are constantly changing, it is important to regularly consult a frequently-updated list or medical reference texts such as the Physicians' Desk Reference may also provide this information. As part of the enrollment/informed consent procedures, the patient will be counseled on the risk of interactions with other agents, and what to do if new medications need to be prescribed or if the patient is considering a new over-the-counter medicine or herbal product. Strong inhibitors and inducers of CYP3A4 are prohibited within 2 weeks before the start of and during treatment. Strong inhibitors and inducers of CYP2C8 should be used with caution; the PI of the study is to be consulted regarding their use * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, or psychiatric illness/social situations that would limit compliance with study requirements * Pregnant women are excluded from this study because of the potential for teratogenic or abortifacient effects. Because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother, breastfeeding should be discontinued during the study and for at least 6 months after last dose of study drugs. These potential risks may also apply to other agents used in this study. Women of child-bearing potential who do not agree to use adequate contraceptive measures during study therapy and for at least 6 months after the completion of study therapy will be excluded. Should a patient become pregnant or suspect she is pregnant while she is participating in this study, the patient should inform the treating physician immediately * Serious or non-healing wound, ulcer or bone fracture * History of abdominal fistula, gastrointestinal perforation or intra-abdominal abscess within 6 months prior to day 1 * Invasive procedures defined as follows: * Major surgical procedure or significant traumatic injury within 28 days prior to day 1 therapy, or open biopsy within 7 days prior to day 1 therapy * Anticipation of need for major surgical procedures during the course of the study * Significant vascular disease (e.g., aortic aneurysm, requiring surgical repair or recent peripheral arterial thrombosis) within 6 months prior to day 1 * Patients with clinically significant cardiovascular disease are excluded * Inadequately controlled hypertension (HTN) (systolic blood pressure \[SBP\] \> 160 mmHg and/or diastolic blood pressure \[DBP\] \> 90 mmHg despite antihypertensive medication) * History of cerebrovascular accident (CVA) within 6 months * Myocardial infarction or unstable angina within 6 months * Serious and inadequately controlled cardiac arrhythmia * Significant vascular disease (e.g. aortic aneurysm, history of aortic dissection) * Clinically significant peripheral vascular disease * Evidence of bleeding diathesis or coagulopathy * Patients with known hypersensitivity to Chinese hamster ovary cell products or other recombinant human antibodies * Current symptom of keratitis or retinopathy at \>= grade 2 * Resting electrocardiogram (ECG) with corrected QT interval by Fridericia (QTcF) \> 470 msec or family history of long QT syndrome * History of bowel obstruction within 28 days from proposed start of treatment * History or evidence of arterial thrombotic or hemorrhagic disorders within 3 months before proposed start of treatment, non-healing wound, ulcer, or bone fracture * Prior use of weekly paclitaxel or bevacizumab in the platinum resistant (disease progression within 6 months of platinum based chemotherapy)/refractory (disease progression during or following the 3 months of the first line platinum based chemotherapy) setting * Known active human immunodeficiency virus (HIV) or hepatitis B or C infection