all it would have taken is for him to have one line about joining the black and that it negated any claim he has to the throne. like, people can argue that he gave up any claim to stark lands and whatnot, but that he coulnd't have renounced his targaryen lineage because he didn't know about it, but at the end of the day he sticks with his "i gave all of that up"
except... people named him king in the north... though they didn't name him that due to his family ties... but again, THAT'S the angle he should be coming from. it doesn't matter if his life is a lie because he renounced whatever life he had.