I think you need to insist on the promised extensive diagnostic report which you wrote right on the invoice was a condition of payment and they never objected to. If they can't provide it, insist they reverse the charge. But I think you're going to have to pay for the battery.
According to M. Wong who is always very knowledgable and helpful, the battery was the wrong battery for the car . . . . and according to your research, the vehicle history shows nothing indicating a battery issue/replacement. Since you bought the car used and you didn't buy it from a BMW dealership, unless you can come up with some evidence of how the wrong battery ended up in the car, I think you're stuck paying for it.
Bottom line based on the info provided in this thread - you don't pay for what they can't document/prove, and they don't pay for what you can't document/prove. From a legal perspective, I think that's the way it would come out in the end.