What is the ruling on a doctor receiving a financial reward—in exchange for following up on kidney transplant cases during and after the operation—when this reward is part of the amount paid to the recipient?

There is no harm in you taking a reward for your assistance in the kidney transplant operation, and you are not obligated to inquire about the donor's intention. If the donor sells his kidney, the sin is on him alone, and the recipient is in a state of necessity, so there is no sin upon him.