Thanks for your replies. So, I finally got this club to fund my first coaching badge in this save.
Out of curiosity, I saved an alternative file and RESIGNED to see if I'd still be enrolled in the course or not. First, the club begged me to stay and offered to increase the transfer budget. I said no and left. To my surprise, though fully unemployed, my profile still showed that I was studying for a license. So, once the club has paid for your tuition, you're in for the license even if you leave the club.
(That would have been very mean on my part. I dropped the alternative save, went back to my original save, and signed a new contract.)