I recently bumped someone’s car while at a stop light. There was maybe a very light scratch on the bumper – if it were my car, I wouldn’t have even noticed or cared. We exchanged information and I asked them to call me with an estimate before going through insurance. They did, and the estimates were all over $1000. So I told them to go through my insurance – and never heard anything else about it, they never followed through. So, based on my very limited experience, I might suggest going through insurance – maybe the hassle will discourage them.
Personally, I agree with msfanster – if it were my car, I would thank you for your honesty and not care about my bumper. Seriously, there are way more important things in life. But I will also say that I have had friends who got screwed over by people in this same situation.