In all honesty - I've not seen a more compelling argument to just go "okay I guess I'll go lend my strength to [insert rival power here]".
Sure, she's the literal king's daughter - and a former member of the hero party from which the kingdom itself claims its founding lineage. But there is a breaking point, and surely even those at the top who are pearl-clutching over "the unladylike"ness of it all, should see that the mere threat of someone of the princess' caliber turning enemy could spell existential doom for this entire place.
If this weren't a "comedy first and foremost" sort of story, anyway. As it is, she'll just Violence her way through it all and win over everyone with the power of muscles and potential yuri and that'll be that.
But the fact that everyone continuously thinks they can "get one over on" the very girl who 1v500'd an enemy army and has done everything she's done at the academy thus far, really makes this one to suspend all belief and higher brain function over.