Radioligand Therapy After PSMA PET Guided External Beam Radiotherapy for Treating Post-Prostatectomy Patients With Biochemically Recurrent Prostate Cancer

Inclusion Criteria: * Adenocarcinoma of the prostate, post radical prostatectomy with detectable prostate specific antigen (PSA) * Clinical PSMA PET/CT obtained, with findings of pelvic uptake only (prostate bed, pelvic lymph node uptake, or both) * Eastern Cooperative Oncology Group (ECOG)/Zubrod performance status of 0-2 * Age over 18 Exclusion Criteria: * Contraindications to radiotherapy (including active inflammatory bowel disease or prior pelvic radiotherapy or prior RLT) * Risk factors for Lu-rhPSMA radioligand therapy (Baseline \>= grade 2 myelosuppression, renal insufficiency \[glomerular filtration rate (GFR) \< 60 mL/min\], or xerostomia) * Definitive findings of systemic metastasis prior imaging (if obtained) or biopsy (if obtained) * Unacceptable medical or radiation safety risk * Unmanageable urinary tract obstruction or hydronephrosis; patients with diagnosed or who are at high risk of urinary retention * GFR \< 60 mL/min or creatinine \> 1.5-fold upper limit of normal (ULN) * Liver enzymes \> 5-fold ULN * Total white cell count less than 2.5 x 10\^9 /L * Platelet count less than 75 x 10\^9 /L * Any baseline grade 2 or above myelosuppression, nephrotoxicity, hepatotoxicity, xerostomia, or gastrointestinal (GI) toxicity * Severe acute co-morbidity, defined as follows: * Unstable angina and/or congestive heart failure requiring hospitalization in the last 3 months * Transmural myocardial infarction within the last 6 months * Acute bacterial or fungal infection requiring intravenous antibiotics at the time of registration * Chronic obstructive pulmonary disease exacerbation or other respiratory illness requiring hospitalization or precluding study therapy at the time of registration * Acquired immune deficiency syndrome (AIDS) based upon current Centers for Disease Control and Prevention (CDC) definition; note, however, that human immunodeficiency virus (HIV) testing is not required for entry into this protocol. The need to exclude patients with AIDS from this protocol is necessary because the treatments involved in this protocol may be significantly immunosuppressive. Protocol-specific requirements may also exclude immunocompromised patients