If you involve your insurance company, you will lose NCB and have the claim recorded on your policy. Do not involve them.
In my view £250 seems a lot for damage done by a trolley and I would also dispute the claim if you were looking for that amount from me. You should get more quotes and don't say to the garage that someone else will be paying otherwise they'll quote another silly amount. You could get the whole car painted for the sort of money that you are talking about.
I imagine if you contact the other person and ask for £50, they will just post you a cheque. If you ask for hundreds they will probably contine to say that it wasn't their fault or that the damage wasn't as bad as to warrant that sort of cost.
You could go down the road of a small claims court action but, in my view, the outcome would not be certain and additional costs will be incurred.