Testing Different Dosing Schedules of the Anti-cancer Drug, Lutetium 177Lu PSMA RLT and Its Effect on Patients With Advanced Prostate Cancer, RECIPROCAL Trial

Inclusion Criteria: * PRE-REGISTRATION (STEP 0): Patients must have histological, pathological, and/or cytological confirmation of prostate adenocarcinoma * PRE-REGISTRATION (STEP 0): Patients must have a positive PSMA PET/CT scan (either gallium Ga 68 gozetotide \[68Ga-PSMA-11\], fluorine F 18 piflufolastat \[18F- DCFPyl\], or fluorine F 18 flotufolastat gallium \[18F-rhPSMA-7.3\]), as defined as uptake greater than liver with no PSMA negative measurable soft tissue disease * PRE-REGISTRATION (STEP 0): PSA greater than 2.0 ng/mL * PRE-REGISTRATION (STEP 0): Patients must have progressive mCRPC. Documented progressive mCRPC will be based on at least 1 of the following criteria: * Serum PSA progression defined as 2 consecutive increases in PSA over a previous reference value measured at least 1 week prior. The minimal start value is 2.0 ng/mL * Soft-tissue progression defined as an increase ≥ 20% in the sum of the diameter (SOD) (short axis for nodal lesions and long axis for non-nodal lesions) of all target lesions based on the smallest SOD since treatment started or the appearance of one or more new lesions * Progression of bone disease: evaluable disease or new bone lesions(s) by bone scan (2+2 Prostate Cancer Clinical Trials Working Group 3 \[PCWG3\] criteria, Scher et al 2016) * PRE-REGISTRATION (STEP 0): Patients must have prior orchiectomy and/or ongoing androgen-deprivation therapy and a castrate level of serum testosterone (\< 50 ng/dL or \< 1.7 nmol/L) * PRE-REGISTRATION (STEP 0): Patients must have received at least one androgen receptor pathway inhibitor (ARPI) (to include either apalutamide, darolutamide, enzalutamide, or abiraterone) \* ARPI must be stopped at least 4 weeks prior to pre-registration * PRE-REGISTRATION (STEP 0): Patients must not have previously received a taxane based chemotherapy regimen for mCRPC. Prior docetaxel for metastatic hormone-sensitive prostate carcinoma (mHSPC) or in the neoadjuvant or adjuvant setting is permitted if completed at least 12 months prior to pre-registration * PRE-REGISTRATION (STEP 0): Patients must have recovered to ≤ grade 2 from all clinically significant toxicities related to prior therapies (i.e. prior chemotherapy, radiation, immunotherapy, etc.) * PRE-REGISTRATION (STEP 0): Patients on a stable bisphosphonate or denosumab regimen for ≥ 30 days prior to pre-registration are eligible * PRE-REGISTRATION (STEP 0): Previous treatment with strontium Sr-89 (strontium-89), samarium Sm-153 (samarium-153), rhenium Re 186 (rhenium-186), rhenium Re 188 (rhenium-188), radium Ra 223 (radium-223) or hemi-body irradiation within 6 months prior to pre-registration is not allowed. Previous PSMA-targeted radioligand therapy is not allowed * PRE-REGISTRATION (STEP 0): Any systemic anti-cancer therapy (e.g. chemotherapy, immunotherapy or biological therapy \[including monoclonal antibodies\]) within 28 days prior to pre-registration is not allowed * PRE-REGISTRATION (STEP 0): Age ≥ 18 years * PRE-REGISTRATION (STEP 0): Eastern Cooperative Oncology Group (ECOG) Performance Status ≤ 2 * PRE-REGISTRATION (STEP 0): Absolute neutrophil count (ANC) ≥ 1,500/mm\^3 * PRE-REGISTRATION (STEP 0): Platelet count ≥ 100,000/mm\^3 * PRE-REGISTRATION (STEP 0): Total bilirubin \< 1.5 x upper limit of normal (ULN) or \< 3 x ULN in patients with Gilbert's syndrome * PRE-REGISTRATION (STEP 0): Creatinine clearance estimated glomerular filtration rate (eGFR) ≥ 40 mL/min/1.73m\^2 using the Modification of Diet in Renal Disease (MDRD) equation * PRE-REGISTRATION (STEP 0): No acute biliary or urinary obstruction * PRE-REGISTRATION (STEP 0): Patients with treated/stable brain metastases are eligible if follow-up brain imaging after central nervous system (CNS)-directed therapy shows no evidence of progression \* Patients with a history of CNS metastases must have received therapy (surgery, radiotherapy, gamma knife) and be neurologically stable, asymptomatic, and not receiving corticosteroids for the purposes of maintaining neurologic integrity. Patients with epidural disease, canal disease and prior cord involvement are eligible if those areas have been treated, are stable, and not neurologically impaired. For patients with parenchymal CNS metastasis (or a history of CNS metastasis), baseline and subsequent radiological imaging must include evaluation of the brain (MRI preferred or CT with contrast) * PRE-REGISTRATION (STEP 0): Patients with known HIV infection on effective anti-retroviral therapy with undetectable viral load within 6 months prior to registration are eligible for this trial * PRE-REGISTRATION (STEP 0): For patients with evidence of chronic hepatitis B virus (HBV) infection, the HBV viral load must be undetectable on suppressive therapy, if indicated * PRE-REGISTRATION (STEP 0): Patients with a history of hepatitis C virus (HCV) infection must have been treated and cured. For patients with HCV infection who are currently on treatment, they are eligible if they have an undetectable HCV viral load * PRE-REGISTRATION (STEP 0): Patients with known history or current symptoms of cardiac disease, or history of treatment with cardiotoxic agents, should have a clinical risk assessment of cardiac function using the New York Heart Association Functional Classification. To be eligible for this trial, patients should be class II or better * PRE-REGISTRATION (STEP 0): No investigational agents within 28 days prior to pre-registration * PRE-REGISTRATION (STEP 0): No other concurrent cytotoxic chemotherapy, immunotherapy, radioligand therapy, or investigational therapy * PRE-REGISTRATION (STEP 0): No known hypersensitivity to the components of the study therapy or its analogs * PRE-REGISTRATION (STEP 0): No transfusion within 30 days of pre-registration * PRE-REGISTRATION (STEP 0): No symptomatic cord compression, or clinical or radiologic findings indicative of impending cord compression * PRE-REGISTRATION (STEP 0): Ability to read and comprehend English or Spanish * REGISTRATION (STEP 1): Completion of 2 doses of 177Lu PSMA RLT * REGISTRATION (STEP 1): PSA decline ≥ 50% between C1 D1 (screening) and C2 D22 +/-3 days * REGISTRATION (STEP 1): ECOG Performance Status ≤ 2 * REGISTRATION (STEP 1): Absolute neutrophil count (ANC) ≥ 1,500/mm\^3 * REGISTRATION (STEP 1): Platelet count ≥ 100,000/mm\^3 * REGISTRATION (STEP 1): Creatinine clearance eGFR ≥ 40 mL/min/1.73m\^2 using the Modification of Diet in Renal Disease (MDRD) equation