To be honest I think that in Raphael's mind, not dealing with things on his own would be more shameful. He's caught up on what his father told him about being a disappointment and he wants to prove himself, even if that means playing dirty (using a real sword) or biting Sophie (seemingly his very last ditch effort to protect Ul, because he wants to prove he CAN protect Ul to his father while maintaining his family's dignity). I mean using a real sword SHOULD be a reason to disqualify him but it isn't bc he is a Delico. Also, Angelico is also from an equally powerful family, so he can't just complain without further unresolved beef being created anyways (which is one of the things that has put Ul in danger already).
Also also, Sophie is in Raphael's eyes the biggest problem right now. Even if he picks a fight with Angelico, which if he won against Sophie he could fight anyways, Sophie still exists as someone Ul wants to get close to (can't happen if Sophie is dead/under control). Right now Raphael is dealing with the cocoon period and his struggle with his father, which have made him emotionally unstable and he wants to be able to fix things on his own and on his terms to prove himself as not a disappointment for his father who he admires.
Don't you just love how they never tell the teachers what's going on? Like u just got fucking stabbed by your fellow student that is controlled by your "rival" and just decided to be quiet???? Why? Do you not have a shit ton of power because of your father??? Is your family name a joke?
And why the fuck do this emotionally constipated ass think the way to make shit better is to enslave/kill (?) Sophie??? Like make it make sense, u wanna pick a bone with someone for fucking everything up go kick Angelicos ass he is the goddamn problem.