177^Lu-PSMA-617 in Combination With Sipuleucel-T for the Treatment of Metastatic Castration-Resistant Prostate Cancer

Inclusion Criteria: * Documented informed consent of the participant and/or legally authorized representative * Assent, when appropriate, will be obtained per institutional guidelines * 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) ≤ 1 * Male * Progressive castration-resistant metastatic prostate cancer with pathologically confirmed adenocarcinoma of the prostate without small cell features * Patients must have either: * Measurable disease * For extranodal (visceral) lesions (e.g. lung, liver, etc.) to be considered measurable, they must be ≥ 10 mm in one dimension, using spiral CT * For lymph nodes to be considered measurable (i.e., target or evaluable lesions), they must be ≥ 20 mm in at least one dimension, using spiral CT * OR non-measurable disease * All other lesions, including small lesions (longest diameter \< 20 mm with conventional techniques or \< 10 mm with spiral CT scan) and truly non-measurable lesions * Lesions that are considered non-measurable include bone lesions (only). Progression on first generation ADT * Patients must have been on androgen deprivation therapy with a gonadotrophin releasing hormone (GnRH) analogue, antagonist, or bilateral orchiectomy (i.e., surgical or medical castration) for at least 3 months prior to study entry and maintain castrate levels of serum testosterone \< 50 ng/dL throughout study participation unless intolerant * Fully recovered from the acute toxic effects (except alopecia) to ≤ grade 1 to prior anti-cancer therapy * Absolute neutrophil count (ANC) ≥ 1,500/mm\^3 (within 10 days prior to day 1 of protocol therapy) * White blood cell (WBC) counts \> 2500/uL (within 10 days prior to day 1 of protocol therapy) * Lymphocyte count ≥ 300/uL (within 10 days prior to day 1 of protocol therapy) * Platelets ≥ 100,000/mm\^3 (within 10 days prior to day 1 of protocol therapy) * Hemoglobin ≥ 9g/dL (within 10 days prior to day 1 of protocol therapy) * NOTE: Red blood cell transfusions are not permitted within 14 days of hemoglobin assessment unless cytopenia is secondary to disease involvement * Total bilirubin ≤ 1.5 x upper limit of normal (ULN) (within 10 days prior to day 1 of protocol therapy) (unless has Gilbert's disease, serum bilirubin level ≤ 3 x ULN) * Aspartate aminotransferase (AST) ≤ 2.5 x ULN (within 10 days prior to day 1 of protocol therapy) * Alanine aminotransferase (ALT) ≤ 3.0 x ULN (within 10 days prior to day 1 of protocol therapy) * Alkaline phosphatase ≤ 3 x ULN (within 10 days prior to day 1 of protocol therapy) (Patients with documented bone metastases, alkaline phosphatase \[ALP\] ≤ 5 x ULN) * Serum creatinine ≤ 1.5 x ULN or creatinine clearance of ≥ 50 mL/min per Cockcroft-Gault formula (within 10 days prior to day 1 of protocol therapy) * If not receiving anticoagulants: International normalized ratio (INR) OR prothrombin (PT) ≤ 1.5 x ULN (within 10 days prior to day 1 of protocol therapy) * If on anticoagulant therapy: PT must be within therapeutic range of intended use of anticoagulants (within 10 days prior to day 1 of protocol therapy) * If not receiving anticoagulants: Activated partial thromboplastin time (aPTT) ≤ 1.3 x ULN (within 10 days prior to day 1 of protocol therapy) * If on anticoagulant therapy: aPTT must be within therapeutic range of intended use of anticoagulants (within 10 days prior to day 1 of protocol therapy) * Seronegative for HIV antigen (Ag)/antibody (Ab) combo, hepatitis C virus (HCV), active hepatitis B virus (HBV) (surface antigen negative), and syphilis (rapid plasma reagin \[RPR\]) (within 10 days prior to day 1 of protocol therapy) * 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 * 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 * For male patients with partners of childbearing potential, agreement (by patient and/or partner) to use highly effective form(s) of contraception or abstain from heterosexual activity for the course of the study through at least 4 months after the last dose of protocol therapy * Childbearing potential defined as not being surgically sterilized (men and women) or have not been free from menses for \> 1 year (women only) Exclusion Criteria: * Any approved or investigational anticancer therapy, including chemotherapy, hormonal therapy (e.g., androgen receptor \[AR\] antagonists, 5 alpha reductase inhibitor, estrogen), or radiotherapy, within 4 weeks prior to initiation of study treatment * Treatment with any of the following medications or interventions within 28 days of registration: * External beam radiation therapy or surgery * Chrysanthemum morifolium/Ganoderma lucidum/Glycyrrhiza glabra/Isatis indigotica/Panax pseudoginseng/Rabdosia rubescens/Scutellaria baicalensis/Serona repens supplement (PC-SPES) (or PC-SPEC) or saw palmetto * Systemic corticosteroids. Use of inhaled, intranasal, and topical steroids is acceptable * Megestrol acetate (Megace®), diethyl stilbestrol (DES), or cyproterone acetate * Ketoconazole * 5-alpha-reductase inhibitors (e.g., finasteride \[Proscar®\], dutasteride \[Avodart®\]) * High dose calcitriol (1,25\[OH\]2 vitamin \[Vit\]D) (i.e., \> 7.0 ug/week) * Prior treatment with 177\^Lu-PSMA-617 and/or sipuleucel-T * Radiation therapy for bone metastasis within 2 weeks or any other radiation therapy within 4 weeks before first dose of study treatment * Known clinically significant liver disease, including active viral, alcoholic, or other hepatitis; cirrhosis; fatty liver; and inherited liver