If I remember correctly they went over this in the manga, concluding that it would be too risky for a long term operation like Strix: even if he were convincing enough to fool the kid whose father he's disguising as, there would be the inevitable risk of said father later learning he'd been seen at events he'd never actually attended, and that could easily lead to even tighter security (including SSS) at the school given all the networking between powerful individuals that seems to go on at Eden