First of all thank you for the donation. If your car was running normal when you brought it in for the sticky throttle and then cleaned it then upon starting it popped and lost compression that's not a coincidence. They must have sprayed a lot of cleaner in the throttle body while it was on the intake and loaded the engine up with all that cleaner and made a big explosion when they tried to start it inside the engine.I would say it would be there cost to repair it. It could have just caused the chain to slip but if the valve were open when the pistons came up it could have damaged valve's as well. Since the shop has already said they won't pay for it then your going to have to take other action's. Here in california we have something called the bureau of automotive repair that is who you would complain to here in my state. You would contact them and tell them what happened and they would contact the shop and be a go between you and the shop. They get a lot more respect and result's then you or I would by ourselves. There must be something like that for your state what state do you live in?There is also the better business bureau you can even file a claim againist a business online even. There is also small claim's court but I would try the two other route's first I think they would be faster result's.