I would just add them as an artist for now.
It's not always possible to credit things perfectly if you only have imperfect information, and in this case they are being treated as their own entity on the poster/ gig details.
Whether you then want to spend your time deciding exactly what kind of school/company they are is up to you
EDIT: If you're not comfortable with that, just leave it as free text. The PLAY situation I think is going to have to stay that way.
And by the way, I believe setlists are very rudimentary at the moment, I'm sure they'll be updated eventually™