Unpopular opinion: don't think "Adelaide" is any less male in his behavior: he saw, knew, aimed, and executed like a guy wooing a girl (and it worked...she's crushing for 5 years). Prince, though, is in denial.
Given that her entire background was found to be falsified, surely they'd suspect it was a foreign power employing a covert agent to break up the engagement to the detriment of the kingdom, and therefore, the engagement would be renewed?
The scandal that got him on house arrest wasn't over whether Adelaide was fake or not, the scandal was over the fact that he was a horndog doing flagrant PDA, groping and sucking the fingers of someone who was not his fiance in full view of the other students of the school who all the children of noble families the royals rely on to run the country.
Adelaide's plot may he been a sting operation, but he disgraced himself of his own volition with his own actions in front of everybody who would be supporting him in the future.