Time travel would be more consistent and easily exploitable since it has less variables but Isekai would be more "fun" even if no cheat powers.
Anyway I don't think it is a missunderstanding, so far it has been very clear that Julia and the Prince are good friends but do not have romantic feelings for one another and, if anything, Ulisses really does seen to like/love Meliarose as he has shown signs of being jealous of Bertram.