Well, I'm all for the place that ripped the seat having to pay for it, but let's acknowledge the occasional difficulty with getting recompensed for a mechanic's screwup.
"Oh, no -- that rip was there. That's an old car. If it wasn't ripped, must have been ready to go. Next person who sat on it would have gone through it. Could have just as easily been you."
Convenient how any car built more than two years ago is old.
Hey, I'm all for giving it a shot, but with the hoops they may make you jump through, you might consider just banging it out on your own.
Buddy of mine recently suggested a tactic for getting money you're owed: Show up filthy and smelling like dog poo -- they'll want to get rid of you as soon as possible, and that probably means paying you. :laugh:
Best of luck -- I'd certainly let them know that you expect to be reimbursed entirely for anything they broke.