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

Inclusion Criteria: 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 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 Total bilirubin =< 1.5 x institutional upper limit of normal (ULN) Aspartate aminotransferase (AST) (serum glutamic-oxaloacetic transaminase [SGOT])/ alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase [SGPT]) =< 3 x institutional ULN Creatinine Cockcroft calculated creatinine clearance of >= 60 mL/min OR Glomerular filtration rate (GFR) of 60 mL/min/1.73 m^2 unless data exists supporting safe use at lower kidney function values, no lower than 30 mL/min/1.73 m^2 Human immunodeficiency virus (HIV)-infected patients on effective anti-retroviral therapy with undetectable viral load within 6 months are eligible for this trial For patients with evidence of chronic hepatitis B virus (HBV) infection, the HBV viral load must be undetectable on suppressive therapy, if indicated Patients with a history of hepatitis C virus (HCV) infection must have been treated and cured. For patients with HCV infection who are currently on treatment, they are eligible if they have an undetectable HCV viral load. Patients with treated brain metastases are eligible if follow-up brain imaging after central nervous system (CNS)-directed therapy shows no evidence of progression Patients with a prior or concurrent malignancy whose natural history or treatment does not have the potential to interfere with the safety or efficacy assessment of the investigational regimen are eligible for this trial Patients should be New York Heart Association Functional Classification of class 2B or better Current disease progression according to PCWG3 criteria Ongoing use of luteinizing hormone-releasing hormone (LHRH) agonists/antagonists will be required (unless prior bilateral orchiectomy or pure neuroendocrine carcinoma histology) to maintain testosterone at castrate levels. Patients with a pure neuroendocrine carcinoma histology do not need to be undergoing LHRH agonist/antagonist therapy No concurrent use of other anti-cancer therapies Pregnancy Precaution: The effects of lutetium lu 177 dotatate on the developing human fetus are unknown. For this reason and because radionuclides are known to be teratogenic, male participants and their female partners must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry and for the duration of study participation. Should a woman become pregnant or suspect she is pregnant while her male partner is participating in this study, she should inform her treating physician immediately. Men treated or enrolled on this protocol must also agree to use adequate contraception prior to the study, for the duration of study participation, and 4 months after completion of lutetium lu 177 dotatate administration. Patients must not donate sperm during the study and for 3 months after the last study drug administration Ability to understand and the willingness to sign a written informed consent document. Participants with impaired decision-making capacity who have a legally-authorized representative (LAR) and/or family member available will also be eligible Patients will undergo a Gallium 68 Dotatate PET scan after enrollment. The Gallium 68 Dotatate PET must be positive to proceed with lutetium Lu 177 dotatate therapy. A positive scan will be defined as at least one lesion with an maximum standardized uptake value (SUVmax) > the average standardized uptake value (SUV) of normal liver. The positive lesion(s) can be in any location (bone metastases or visceral metastases). Patients with only bone metastases will be allowed Exclusion Criteria: Patients who are receiving any other investigational agents History of allergic reactions attributed to compounds of similar chemical or biologic composition to Lutetium Lu 177 dotatate As per the Food and Drug Administration (FDA) package insert for Lutetium Lu 177 dotatate, use of long-acting somatostatin analogs (e.g., long-acting octreotide) is prohibited within 4 weeks prior to initiating Lutetium Lu 177 dotatate and during treatment. Use of short-acting somatostatin analogs is prohibited within 24 hours prior to initiating Lutetium Lu 177 dotatate and during treatment. Long-acting somatostatin analogs or short-acting somatostatin analogs will be allowed if the patient has a history of carcinoid syndrome and requires long-acting or short-acting somatostatin analogs for the control of his functional syndrome Patients with uncontrolled intercurrent illness Any of the following within 6 months before starting treatment: stroke, myocardial infarction, severe/unstable angina pectoris, coronary/peripheral artery bypass graft; congestive heart failure New York Heart Association (NYHA) Class III or IV Uncontrolled hypertension as indicated by a systolic blood pressure >= 160 mmHg or diastolic blood pressure >= 100 mmHg at screening