So I'm thinking the obvious fallout, regardless of the solution at the academy is that a conniving politician, likely the one that somehow knows of the investigation, will accuse the Queen of inadequacy. As grounds for her removal, and her heirs, anyone she's chosen will also be deemed not worthy of the throne. The way out of this is to provide evidence that the politician stirring shit up was involved with the plot from the start. It'd show that, yes, he can shine light on the Queen's inadequacy but at the cost of his own leadership. Why would other nobles follow a man that willingly put their own at risk?
The good endgame, the ideal one, would be that the prince comes out of this having shown resolve and leadership. He would be the one to expose the shit-stirrers, thus showing that he does not share the inadequacies of the Queen while also showing his intelligence and skill at political maneuvering. Basically this incident would steel his resolve and result in an actually good King. Not a manipulable one (an ideal King for the nobility), but a competent one in his own right.
Sadly, that's highly unlikely. Unless he grows a pair in the next chapter, MC will do all the work and just say "oh the Prince got the answers". It'd be unfavorable to the nobility, to the royal family, and to the people at large.