An extreme version of this is: What should the German health service do if someone says they are willing to donate a kidney as long as it doesn’t go to a Jew?
On the one hand, nobody is forced to donate a kidney and by forbidding this we’re making things worse for an innocent patient. On the other hand, it can be seen as the state sanctioning this kind of discrimination.
No. Organ “donation” after death should be compulsory. For living donors there should be a publicly funded bounty system where you either take the money or not. Donors and recipients don’t get to be picky.