Okay, hang on my dude, were you just going to say "no" if he just said "yeah, after you teach me, I'm going to go back to help my dad fight?" like an honest and normal 11 year old?
Like you can't exactly just send him back... It's obvious what he was sent south to avoid being caught up in.
(There are third options here, mind. like It'd definitely be within the realm of respecting his dad's stated wishes if the headmaster just made sure there was a place he could live in the south comfortably—that's basically what was originally asked for)