Small damage blown up to Â?2500 claim by the other driver

My parking sensor stopped working and while slowly reversing, I bumped into another car. At the time the owner said that it was only a scratch and wanted to make no claims. No photos were taken. My sensors were replaced by the company who fitted them and the small damage on my bumper was mended at their expense. However, the owner of the other car took her Mini to  BMW garage where they came up with a long list of damages and a Â?2500 bill. I don't think I caused all the damage, as the Mini has other marks on its bumper where I could not have caused damage, so I think they are just trying it on. The owner isn't willing to take the car to another garage appointed by me for a second opinion but demands payment.

As I don't have no claims bonus protection on my otherwise fully comprehensive car insurance, I don't want to claim for a minor damage that is now blown up and was caused by a faulty part. The sensor fitting company also rejected to deal with the Mini's damages.

Any help regarding how best I could go about it to avoid paying for damage I didn't cause and to avoid claiming it on my insurance would be greatly appreciated.