Can stage III bladder cancer patients still get a transplant?

No. I don't think so, but i would check with your transplant coordinator. The problem is that giving immunosuppression to someone with an advanced malignancy is dangerous.
See below. Transplant candidates are screened for cancer as part of the assessment process. For potential transplant recipients with a prior history of cancer, waiting periods are ~2–5 years after remission, depending on cancer type and stage. Kidney transplant recipients have >2x higher risk of developing or dying from cancer than. The higher risk is mulifactorial: immunosuppression, onogenic viruses, etc. .