Autologous CAR T Cells Targeting GPC3 (RPCAR01) for the Treatment of Advanced or Metastatic GPC3 Expressing Hepatocellular Carcinoma

Inclusion Criteria: * ≥ 18 years of age * Pathologically confirmed diagnosis of hepatocellular carcinoma. Mixed hepatocellular cholangiocarcinoma histology will be excluded in this trial * Must have received at least 2 recommended standard of care lines of therapy for HCC which include checkpoint inhibition and a tyrosine kinase inhibitor such as lenvatinib * Tissue confirmation of expression of GPC3 with immunohistochemistry (IHC) on archival tissue. ≥ 50% of tumor cells should be IHC 1+ or greater GPC3 intensity * Presence of measurable disease, with Response Evaluation Criteria in Solid Tumors (RECIST) 1.1 at the time of intervention consent. Previously treated lesions are acceptable as long as there is a new confirmed measurable component * Eastern Cooperative Oncology Group (ECOG) performance status score of 0-1 * Life expectancy of at least 3 months * Patients with chronic hepatitis B virus (HBV) infection with active disease who meet the criteria for anti HBV therapy should be on a suppressive antiviral therapy prior to initiation of therapy * Patients with a history of hepatitis C virus (HCV) infection should have completed curative antiviral treatment and have a HCV viral load below the limit of quantification * Patients with HIV should have CD4+ T-cell (CD4+) counts ≥ 350 cells/µL within 3 months of study enrollment * Leukocytes ≥ 3,000/mcL * Absolute neutrophil count ≥ 1,500/mcL * Platelets ≥ 65,000/mcL * Total bilirubin ≤ 1.5 x 1.3 mg/dL * Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamic pyruvic transaminase \[SGPT\]) ≤ 3 × institutional upper limit of normal (ULN) * Creatinine clearance ≥ 50 mL/min (Cockroft-Gault) * Participants of child-bearing potential must agree to use adequate contraceptive methods (e.g., hormonal or barrier method of birth control; abstinence) prior to study entry. Should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately * Participant must understand the investigational nature of this study and sign an Independent Ethics Committee/Institutional Review Board approved written informed consent form prior to receiving any study related procedure Exclusion Criteria: * Patients eligible for curative surgical or locoregional treatment options at the time of screening will not be included in the clinical trial. * Patients with underlying Child-Pugh B or C liver cirrhosis (score of 7 or over) * Patient does not have archival tumor tissue available for GPC3 testing * Patients with a recent history of ongoing active bleeding * Patients who have a current medical history of alcohol abuse * Patient with a history of grade 2 or greater immune mediated toxicities of a major organ or a history of autoimmune hepatitis, pneumonitis or myocarditis * Participants who have had chemotherapy or radiotherapy within 2 weeks (6 weeks for nitrosoureas or mitomycin C) prior to entering the study or those who have not recovered from adverse events due to agents administered more than 4 weeks earlier. A washout period of 2 weeks is required prior to leukapheresis * Participants with known leptomeningeal disease or brain metastases should be excluded from this clinical trial because of their poor prognosis and because they often develop progressive neurologic dysfunction that would confound the evaluation of neurologic and other adverse events * Concomitant systemic glucocorticoid use at a dose equivalent to \> 10 mg daily prednisone at the time of leukapheresis and/or within 4 weeks of CAR T infusion * Clinical or radiographic evidence of bowel obstruction or need for parenteral hydration and/or nutrition * Active autoimmune disease * Prior history of seizure disorder * Pregnancy or breast-feeding female participants * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements * Unwilling or unable to follow protocol requirements * Any condition which in the investigator's opinion deems the participant an unsuitable candidate to receive study drug