I think one thing you can try doing is apply for a Study Permit. Sometime you may be lucky to get it providing that you can show CIC proof of your intentions of attending school.
I am not 100% sure of this but I also think you are allowed to study in Canada for a period of 6 months without any legal authorizations by CIC.
Maybe someone can give you more information on this. It's still worth trying for sure
I was granted a study permit in Canada after my Refugee, PRRA, and H&C applications were rejected. Mind you, I still have to leave when CBSA decided to remove me. The pro to this was I got to study for about 11 months before having to leave Canada.
Immigration Canada is a very strange system and sometimes the things we think wouldn't work for us end up being reality.
Good luck!