I do not need spells to detect that someone is an ass, same for the cleric.
That prince has a reputation and is known throughout his kingdom, so she already knows before she even met him. No need to detect anything
Ultimately you should not expect your average adventure to go as intended when the party you decided to run it for is so outside of any norm… prepare for a lot of making things up on the fly and getting derailed, pretty much what the DM was asking for.
I’d say cut your losses and go freeform from wherever you are now. There is no way you run that adventure, come up with one that fits the group instead