Interesting that they're putting this on you to do. Talk to your donor advocate at the transplant center. Most likely someone new in a role doesn't know that services for you pre-, during, and post-surgery all get billed to the recipient's Medicare A and/or B coverage, if any, or insurance company otherwise. If you're getting bills, or, like this, the implication that you have to do anything to avoid being billed, the transplant center admin needs to know and resolve.
I got a dunning call nearly two years after I donated about an unpaid anesthesia charge that I'd never seen. The hapless collection agent, a tough talking guy, burst into tears when he found out I was a living donor, apologized, and told me he'd track it back to be coded and billed properly. His daughter was on dialysis and listed for a deceased donor transplant. Best wishes to you and your recipient!