Need advice please

A few days ago, I (gently) bumped into another car's bumper. It was stop & go traffic jam and the speed was minimal, perhaps 2 mph or less. We pulled over to assess damage. It was dark but his car's bumper looked absolutely normal to me. No dents or scratches. I took photos. No police was called. No damage to my car either. The other driver appeared hostile and said that he will nevertheless get it assessed professionally and send me an estimate.
In any case, today he called me to say that the bumper needs to be re-sprayed and it would cost around £450.
Since this is the first incident, I am not sure whether it would be better to send him the cheque or do it through my insurance.
My worry is that he might be overstating the damage (I could not detect any scratches!). Plus a standard re-spray seems to cost about half the price from a google search. Also what would stop him going for more from my insurance after he receives my cheque?
What can I do to make sure this matter is treated fairly?
Should I get my insurance involved or settle privately? Would it not be better to have my insurance assess the damage? (is it even possible?)
I think my excess is £600 with my co and I have 3 years no claims and this is the first incident.
Advice would be much appreciated.
thanks