Just curious - did they bill you for the airplane retrieval? If so, all the costs or just some?
If it's X then it will be interesting. A few years ago (story is somewhere on POA) 2 airplanes (me and another) flew over to Goodland for lunch and a student pilot had serious mechanical difficulties, and the rental/school demanded that the student pay for everything and would be reimbursed, or else the school would charge for the retrieval.
As I said, I've seen the school's contract, and the reimbursement was credit for lessons and rental, not cash. So OP did right by demanding the school deal with the shop. I'd be interested in what the school ended up charging the OP, too.