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

Inclusion Criteria: 1. Subjects must have histologically or cytologically confirmed 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. 2. Participants must have a PSMA-positive 18F-rhPSMA-7.3 performed within 12 weeks from C1D1 with ≥1 site with SUVmax ≥10) mCRPC with progression on prior novel hormonal agent to include at least one of the following: 1. PSMA SUV mean \<10 2. ≥1 visceral metastasis 3. ≥5 bone metastases OR two of the following 1. TP53 2. PTEN 3. RB1 mutation. 3. Age \> 18 years. 4. ECOG performance status of 0 to 2. 5. Subjects 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: Subjects 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 subjects 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 6. Subjects 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. 7. Subjects 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 participants must also practice approved forms of birth control 8. Subjects must have the ability to understand and the willingness to sign a written informed consent form (ICF). 9. Members of all races and ethnic groups are eligible for this trial. Exclusion Criteria: 1. Evidence of hormone-sensitive prostate cancer (HSPC) 2. Evidence of small cell prostate cancer 3. Subjects receiving any other investigational agents. 4. 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. 5. Subjects with brain metastases/central nervous system (CNS) disease that are treated prior to enrollment will be allowed in this clinical trial. 6. Known or suspected significant hypersensitivity to any components of the formulation used for Cabazitaxel, carboplatin or 177Lu-PSMA-617. 7. 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. 8. Prior treatment toxicities not resolved to ≤ Grade 2 according to NCI CTCAE Version 5.0