Can we start a rebellion already, for fuck sake both of the candidates for future king/emperor is a fucking asshat.
You've forgotten the third, fourth, and fifth in line to the throne: Cyan, his uncle, and his spunky cousin. As long as there are some good heirs, you just need to oust the two nasty incompetent princes and the emperor (or you can just wait for him to die, which ever works). Since the two princes are busy committing felonies (to wit, rape and black magic) and this is a European-based setting where the author actually bothered to do at least a little research, the royal family should be beholden to the law as much as anyone else. In conclusion, the princes will lose their rights of succession once their crimes are discovered. As for the emperor, he's mostly unpleasant and a bad father, neither of which is criminal or bad for the nation by itself, so waiting for him to pass on is probably the safest bet. That's a
lot less bloodshed, not to mention quicker.