PSCA-Targeting CAR-T Cells Plus or Minus Radiation for the Treatment of Patients With PSCA+ Metastatic Castration-Resistant Prostate Cancer

Inclusion Criteria: Documented informed consent of the participant and/or legally authorized representative (brown) Assent, when appropriate, will be obtained per institutional guidelines Note: For research participants who do not speak English, a short form consent may be used with a City of Hope (COH) certified interpreter/translator to proceed with screening and leukapheresis, while the request for a translated main consent is processed. However, the research participant is allowed to proceed with lymphodepletion and CAR T cell infusion only after the translated main consent form is signed Agreement to allow the use of archival tissue from diagnostic tumor biopsies If unavailable, exceptions may be granted with study principal investigator (PI) approval Age: >= 18 years Eastern Cooperative Oncology Group (ECOG) performance status 0-2 or Karnofsky Performance Status (KPS) >= 70% Documented castration resistant prostate cancer (mCRPC) (Note: castration will be defined by a testosterone < 50 ng/dL achieved by orchiectomy or luteinizing hormone-releasing hormone [LHRH] agonist/antagonist therapy) Documented PSCA+ tumor expression as evaluated by the COH Pathology Clinical Trials Specimen Qualification Laboratory (CTSQL) Fresh or archival biopsy samples may be tested for PSCA expression during screening for eligibility purposes. The results from soft tissue biopsies will be used to confirm eligibility for participants who have a soft-tissue lesion biopsy obtained, but bone biopsy staining results will not impact eligibility since immunohistochemistry (IHC) staining for PSCA has not been optimized in bone specimens. Subjects who undergo bone biopsy on study will be qualified based on the archival tissue result Progression of disease manifest by one of the following means during treatment with at least one advanced androgen targeted therapy (e.g., abiraterone or enzalutamide): Rising prostate specific antigen (PSA) documented on 2 occasions at least 7 days apart, with absolute increase > 2 ng/dL despite testosterone < 50 OR Radiographic evidence of new metastatic foci on CT or bone scan, or soft tissue progression by Response Evaluation Criteria in Solid Tumors (RECIST) For treatment plan 2, subjects must have at least one and up to 3 metastatic lesions which have not previously been radiated and which is safe for treatment with radiation 16 gray (Gy) in 2 fractions Fully recovered from the acute toxic effects (except alopecia) to =< grade 1 to prior anti-cancer therapy If there has been prior chemotherapy, at least 2 weeks must have elapsed prior to leukapheresis Prior radiotherapy is allowed provided it was not administered to the only evaluable site of disease and was completed > 14 days prior to leukapheresis No known contraindications to leukapheresis, steroids or tocilizumab Absolute neutrophil count (ANC) >= 1,000/mm^3 (within 42 days prior to enrollment) NOTE: Growth factor is not permitted within 14 days of ANC assessment Platelets >= 100,000/mm^3 (within 42 days prior to enrollment) NOTE: Platelet transfusions are not permitted within 14 days of platelet assessment Total serum bilirubin =< 2.0 mg/dL (within 42 days prior to enrollment) Patients with Gilbert syndrome may be included if their total bilirubin is =< 3.0 x upper limit of normal (ULN) and direct bilirubin =< 1.5 x ULN Aspartate aminotransferase (AST) =< 2.5 x ULN (within 42 days prior to enrollment) Alanine aminotransferase (ALT) =< 2.5 x ULN (within 42 days prior to enrollment) Creatinine clearance of >= 50 mL/min per 24 hour urine test or the Cockcroft-Gault formula (within 42 days prior to enrollment) Corrected QT interval (QTc) =< 480 ms Note: to be performed within 28 days prior to day 1 of protocol therapy Cardiac function (12 lead- electrocardiogram [ECG]) without acute abnormalities requiring investigation or intervention (within 42 days prior to enrollment) Seronegative for human immunodeficiency virus (HIV) antigen (Ag)/antibody (Ab) combo, hepatitis C virus (HCV), active hepatitis B virus (HBV) (surface antigen negative), and syphilis (rapid plasma reagin [RPR]) If positive, hepatitis C ribonucleic acid (RNA) quantitation must be performed OR If seropositive for HIV, HCV or HBV, nucleic acid quantitation must be performed. Viral load must be undetectable Note infectious disease testing to be performed within 28 days prior to day 1 of protocol therapy Meets other institutional and federal requirements for infectious disease titer requirements Note Infectious disease testing to be performed within 28 days prior to day 1 of protocol therapy Agreement by males of childbearing potential to use an effective method of birth control or abstain from heterosexual activity for the course of the study through at least 3 months after the last dose of protocol therapy Childbearing potential defined as not being surgically sterilized Exclusion Criteria: Concurrent use of systemic steroids or chronic use of immunosuppressant medications. Recent or current use of inhaled steroids is not exclusionary. Physiologic replacement of steroids (prednisone =< 7.5 mg /day, or hydrocortisone =< 20 mg /day) is allowed Subjects with clinically significant arrhythmia or arrhythmias not stable on medical management within two weeks of screening Subjects with a known history or prior diagnosis of optic neuritis or other immunologic or inflammatory disease affecting the central nervous system, including seizure disorder History of allergic reactions attributed to compounds of similar chemical or biologic composition to study agent Known bleeding disorders (e.g., von Willebrand's disease) or hemophilia History of stroke or intracranial hemorrhage within 6 months prior to screening History of other malignancies, except for malignancy surgically resected (or treated with other modalities) with curative intent, basal cell carcinoma of the skin or localized squamous cell carcinoma of the skin; non-muscle invasive bladder cancer; malignancy treated with curative intent with no known active disease present for >= 3 years Clinically significant uncontrolled illness Active infection requiring antibiotics Known history of immunodeficiency virus (HIV) or hepatitis B or hepatitis C infection Any other condition that would, in the Investigator's judgment, contraindicate the patient's participation in the clinical study due to safety concerns with clinical study procedures Prospective participants who, in the opinion of the investigator, may not be able to comply with all study procedures (including compliance issues related to feasibility/logistics)