I would also suggest you contact the transplant center and/or insurance company to find out the answers.
In my case, my recipient's insurance covered the cost of the donation surgery and hospitalization for me. I had no complications, but I'm guessing that if I had any complications immediately due to surgery that required other treatment, longer than usual stay, etc, that would have been covered as well. Once I was discharged from the hospital, my own insurance would pay for any other care needed.
I would guess that once you are discharged from the initial surgery, your own insurance would have to cover, so depending on how long you stay in the US before returning home, that's the period you need to be concerned about. I would also find out if the refusal to cover voluntary medical procedures means they won't cover any complications from said procedures, because they might consider those two separate things. So for example, if you get an infection post-discharge and have to return to the hospital for treatment, will both the national health insurance and any travel insurance refuse to pay for that?
But again, this is all speculation and only the insurance company can give you the information you need. The transplant center can most likely also help, since you can be sure they want to get paid so they know what is covered.