Testing the Safety and Effectiveness of Radiation-based Treatment (Lutetium Lu 177 Dotatate) for Metastatic Prostate Cancer That Has Neuroendocrine Cells

Inclusion Criteria: * PRE-REGISTRATION ELIGIBILITY * Patients must have metastatic prostate cancer with neuroendocrine differentiation, as determined by at least one of the following: * Histologically confirmed small cell or neuroendocrine cancer from a primary prostate or metastatic biopsy. Neuroendocrine prostate cancer includes mixed small cell with adenocarcinoma histology, as well as small or large cells with positive neuroendocrine markers (e.g., chromogranin or synaptophysin) * Prostate adenocarcinoma with molecular features of neuroendocrine differentiated cancer (e.g., 2 of the following 3: PTEN, TP53, or RB loss) * Progression of visceral metastases in the absence of PSA progression * Serum chromogranin A \> 5x normal limit, or neuron-specific enolase \> 2x normal NOTE: Both patients who have had prior cytotoxic chemotherapy and patients who have never had cytotoxic chemotherapy for prostate cancer will be allowed * Age \>= 18 years. Prostate cancer is typically a disease of older men, with the average age at diagnosis being 65 years. Consequently, because the research topic is not relevant to children, no children will be included in this study. There is no upper limit to the age of participants eligible for this study * Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 (Karnofsky \>= 60%) * Absolute neutrophil count (ANC) \>= 1,500/mcL * Platelets \>= 100,000/mcL * Hemoglobin \>= 8 g/dL, prior to each dose of lutetium lu 177 dotatate * T…