When I first read about R+L = J it talked about how it instantly made Jon's claim to the throne potentially stronger than Dany's, but you sound like you know what you're talking about here, so I must defer. It would definitely become a very interesting source of conflict, and I can't imagine a "Dany and Jon happily ever after, uniting to triumph over all odds and evils" storyline to finish the show. That'd be disappointing, really.
Bastards don't have claims to thrones. He has zero claim. If we pretend his parents were married, no one on earth knows or thinks they were married except maybe Bran, and like I said it probably wasn't a legitimate marriage anyway. Neither side wanted it. Robert would have killed both of them if he thought his woman was in the relationship willingly (married or not), and the mad king probably would have too.
I don't think there will be any conflict, unless it's from Dany being paranoid, because Jon is a very reluctant ruler in the first place. (I think she will be thrilled to have a living relative). King of the North is a bit different since it's a brand new title, and also because the Starks were decimated. At one point everyone thought there was no male heir to the throne. They also thought Sansa was lost to the Lannisters forever and Arya was dead. That left literally no one from the strongest house in the North except Jon. So he, a bastards, becoming their ruler was only because of extremely odd circumstances. Also, when Stannis offered to make Jon a legitimate son of Ned, he turned him down.
Agreeing to rule the north was a big leap, and I think Sansa will end up doing that even if Jon survives. Jon wouldn't accept being king of everything. What he would accept, IMO, is ruling the north for Dany, or giving Sansa a strong recommendation and helping bring them to some kind of agreement.
That ending wouldn't be disappointing at all, given all the sh#t everyone's gone through.