Not a lawyer, but personally I don't think you have a case as you are missing some very basic fundamentals:
a.) You have not proven what is wrong with the vehicle. All's you are reporting are the symptoms, and then somehow leaped to a conclusion of what the problem MUST be. There are many things that could cause a rough running engine. You don't know definitively what is wrong yet. You don't know whether it IS a camshaft bearing, an engine computer, a head gasket, a fuel injector, a bent rod, a valve, a crankshaft sensor, ayada yada yada.
b.) Since you don't definitively know what is wrong (the root cause of the engine running rough), you also don't know (or have proved) what the problem is, and then proven what caused the problem. Again, you have some assumptions, but you have no proof.
c.) Since you don't know the engine problem, you also don't know what repair options and costs are required to fix the problem. You have some assumptions as to what it would take. But lets say you have a bad valve, the cost to replace/resurface a valve, is different than the costs to repair a main crank journal.
I think you're going to have to spend some money to have the engine torn apart and analyzed, and give the oil change shop the capability to observe and inspect. It unfortunately might be throw away money, if it doesn't show you what you need to prove your current theory.
I could go on, but I think you're missing the very basics to even start.