Help! Authorization needed for part-time internship

back in late October I managed to land a part time internship. I've been there twice for some training already, but the problem is that I'm an international student and they insist that they won't let me work unpaid and that I will need authorization. Problem is, my school doesn't let undergrads use CPT so I'd have to apply for OPT to start working there, and OPT takes anywhere from 2-3 months to process so I wouldn't be able to start this semester.

I think that the experience would be valuable and I want to list it on my resume, and it's annoying that this delay is preventing me from working while i know other people have worked unpaid without any problems. Given that it's likely I'll be authorized and have started by the time background check rolls around (if I land an SA), would I be able to list the internship on my resume?

also, has anyone been in this situation before and managed to work around it without having to apply for OPT? I'm a Canadian citizen, if it helps