[Belegorn]

I would think the trouble with an immortal king of Gondor would not be for the king's own cravings for immortality as he already is immortal, but his mortal descendants who can never claim the throne and die as their ancestor who was able to choose an immortal life cannot share that choice with them. However, I'm not sure he would be fit to rule the Men of Gondor because he would no longer be a Man. So maybe if one of his other siblings choice mortality they would be that ruler and not himself. Or the reign would fall to one of the Dunedain close to the king's house.