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 G…