This phase I trial studies the safety, side effects, and treatment cycle 1 best dose of Lu-177-PSMA-617 in patients with prostate cancer that keeps growing even when the amount of testosterone in the body is reduced to very low levels (castration-resistant) and has spread from where it first started (primary site) to other places in the body (metastatic). Lu-177-PSMA-617 is a radioactive drug. It binds to a protein called prostate-specific membrane antigen (PSMA), which is found on prostate cancer tumor cells. Lu-177-PSMA-617 gives off radiation that may kill these tumor cells. It is a type of radioconjugate. Lu-177-PSMA-617 is currently used in a series of 6 intravenous infusions of the standard dosage, each separated by 6 weeks from the previous infusion. Investigators have observed that the first therapy administration (cycle 1) delivers better radiation treatment to the cancer than each of the following 5 infusions. Giving an increased dosage of Lu-177-PSMA-617 in treatment cycle 1 may have a better effect on metastatic castration-resistant prostate cancer than the standard dosage. The study does not change the total cumulative activity from what is used in the standard dosage treatment but gives an increased dosage in cycle 1 and reduces the total number of cycles.