I really don't see any reason for him to give up on being the runestone heir since as heir he would have a safe place and an army and as for the other lords what will they be able to do he has two dragons an army and starks support what basically means the north and i doubt the king would do anything against him honestly he should be more worried about his mother and the war that is going to happen.
I hope he doesn't give up being heir since it doesn't make sense for him to give up his inheritance just because he's worried about what others will think let's be honest he has a dragon that already gives a huge advantage in battles and he still has the the fact that it won't be long before a war breaks out so the opinion of the other lords should be the least of your worries as I really don't see a way he won't have to fight this war as he has dragons and it's almost impossible for him to run away to fight I'm really curious to know what will happen