Hi, you most certainly have a case and yes if you can run to it get a solicitor involved even if its only advice, it is indeed their problem as the car was not fit for purpose, as for Audi approved I wouldn't give an old snotty tissue for their idea of checks the AA are about the most honest and precise checkers of car faults as they have no allegiance to any dealer and by the sound of it a trained Ape would have spotted that fault, I have personally witnessed the obscene lack of attention to detail with their service plans, go in hard and fast while you still have the whip hand time wise and don't let them string you along, the uneconomical repair bullocks won't hold water as wether or not under sold won't come into it as they were happy with that price, they sold you a car that is defective and not as advertised so taking in your after purchase fiscal outlay you obviously intended to keep the car and want it put back to as advertised condition, they are not known as Audi main Stealers for nothing.
Steve.