I laughed at the beer comment, although I am not into them myself this evening.

I might have been a bit remiss in regard to what I did learn reviewing what the AMA offered in regard to ethics.
Personally, I don't think he should have gotten the transplant, if dialysis was still available, if he was here illegally and if it came from the "US Donor Pool", which was supported by
some of the folks at the AMA. However, there was no clear indication as where the man in question received his transplanted kidney, and where he may sit on the list as far as transplants go can be (not always) influenced by whether or not the transplant comes from a dead donor or a living donor (in some cases Americans receive better standing where dead donors are concerned).
The objections to doing the transplant on Illegal Foreign Nationals (like mine), were centered on reason and the realization that should deportation come into play the organ could be considered to be a "waste". The opposing opinions consisted of a rather "blank check" approach to the idea nothing really mattered other than giving the person the kidney.
Also, contrary to popular belief in some circles, surgeries where Illegal Foreign Nationals receive transplants, are paid for by a combination of charities (when available) combined with either federal or state funds as afforded through either Medicaid or provisions in the ACA (and it is a specific provision). In fact, the AMA has a term for them and they are called "Riders".
However, his legality as the AMA currently indicated would not affect whether or not he was in the system. Simply reporting to the hospital for care would start the process where everything would be handled and Illegal Foreign Nationals do get transplants.
My best solution in the circumstance you mentioned would be more direct, and I would have made sure the recipient had a doctor's note (possibly a prescription which from reading what I did would most likely be what they were referring to) and we would not even be having this conversation, but no one asked me and that makes way too much sense while it would not support a lot of political nonsense.
Maybe that "bunny trail was a tad bit longer than I initially let on, but I tried to condense the issue and information without adding a lot of opinion (except when I indicated it was my opinion).