Dilemma - to pay or not to pay?

Did you sign a work agreement prior to their investigation of the problem. If so you are probably bound by the agreement. If an agreement is in place the shop can doc your credit, take you to court at your expense, turn you over to collection, and depending on the state perhaps put a lean on your registration. That aside: Think about out shop. A technician was assigned to your job, perhaps the owner, and hours were spent searching out a solution you admit was really difficult to find. Perhaps with a few more hours of searching they could have found the solution...at an even greater expense.
Another point, which really should not have any bearing on the outcome, but I'll mention it anyway: Can you afford the charge? Personally, if it is just a matter of principle...pay the guy, he worked for it.
JMHO