Kit already suggested in an interview that Jon is not cool with incest. I think the most immediate effect of Jon learning his truth will be an immediate relationship freeze, where he will wonder if he can continue this. (Eventually he'll be cool with it)
But he can renounce any rightful claim and I don't see any reason why he will suddenly feel it's not Dany's Throne. And I don't see why Dany would have some fatal concern about Jon or his super clear lack of ambition.
So...that's either a puzzle there, or not. Maybe the simple solution is that Dany graciously (after much rumination & counsel) settles for Queen to quell possible dissatisfaction with her claim, confident her reform policies will be implemented by Jon (as her frontman basically).