Cabazitaxel +/- Carboplatin vs 177Lu-PSMA-617 in Metastatic Castrate-resistant Prostate Cancer

Inclusion Criteria: * Participants must have histologically or cytologically confirmed adenocarcinoma of prostate * Evidence of metastatic castrate-resistant prostate cancer that has previously been treated with an androgen receptor pathway inhibitor. Prior docetaxel exposure is recommended but not mandatory. Tissue is not mandatory, but a pathologic report is required at time of enrollment. * Patients must have a PSMA-positive 18F-rhPSMA-7.3 performed within 12 weeks from C1D1 with ≥1 site with SUVmax ≥10. An alternative PSMA PET tracer is permitted at baseline if performed within 8 weeks prior to randomization. * Eligible patients have evidence of mCRPC who have progressed on prior novel hormonal agent(s) to include at least one of the following: * Baseline PSMA SUVmean \<10 OR * ≥1 visceral metastasis OR * ≥5 bone metastases OR one of the following (using Next Generation Sequencing on file within 5 years) * TP53 * PTEN * mutation. * Age \> 18 years. * ECOG performance status of 0 to 2. * Participants must have adequate organ and marrow function as defined below to be suitable for the randomized treatment outlined in this * Absolute neutrophil count \>1000/μL; platelet count \>90 000/μL; hemoglobin \>8.5 g/dL) at screening. * Note: Participants must not have received any growth factors within 7 days or blood transfusions within 14 days prior to the hematologic laboratory values obtained at screening). * Total bilirubin (TBIL) \<2.5 × the upper limit of normal (ULN) at screening, except participants with documented Gilbert syndrome who must have a TBIL \<3 mg/dL * Alanine aminotransferase (ALT) and aspartate aminotransferase (AST) \< 5 ULN at screening * Creatinine clearance ≥40 mL/min and/or estimated glomerular filtration rate (eGFR) ≥30 * Albumin \>30 g/L (3.0 g/dL) at screening * Participants receiving bisphosphonates or other approved bone-targeting therapy (e.g., denosumab) must be on a stable dose for at least 14 days before the start of study treatment. * Participants of child-producing potential agree to use highly effective contraceptive methods (i.e., barrier contraception measures such as a male condom with spermicide during intercourse) and avoid sperm donation during the study treatment and for 3 months after the last dose of study treatment. A man is considered to be of child-producing potential, unless he has had a bilateral vasectomy with documented aspermia or a bilateral orchiectomy. Partners of patients must also practice approved forms of birth control * Participants must have the ability to understand and the willingness to sign a written informed consent form (ICF). * Members of all races and ethnic groups are eligible for this trial Exclusion Criteria: * Evidence of hormone-sensitive prostate cancer (HSPC) * Evidence of small cell prostate cancer * Participants receiving any other investigational agents. * Diagnosis of another clinically significant malignancy within the previous 2 years other than curatively treated non-melanomatous skin cancer or superficial urothelial carcinoma and other in situ or noninvasive malignancies, as determined by the PI or Co-PI. * Participants with brain metastases/central nervous system (CNS) disease that are treated prior to enrollment will be allowed in this clinical trial. * Known or suspected significant hypersensitivity to any components of the formulation used for Cabazitaxel, carboplatin or 177Lu-PSMA-617. * Uncontrolled intercurrent illness, including but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations considered by the Investigator to limit compliance with study requirements. * Prior treatment toxicities not resolved to ≤ Grade 2 according to NCI CTCAE Version 5.0