Best bet in my opinion is to take it to a few local body shops (not the main insurance ones!).
Discuss what you want doing, and see what price you get.
You are in for about £500 for a repair of the door and rear panel if not to bad.
Some may take on using a used door, but rear panel still will need re-spraying, so will not save much.
I got estimates for £800 for my TF which I was selling for £1300. Guy who did the damage (a bricklayer who droped bricks on it) didnt want to pay that, so handed it to his insurance. They got a bit snotty with me, wanting deductions for wear and tear and didnt have a policy of making it better than before the accident!, so I handed them over to my insurance company, who paid £1100, and also claimed my £100 excess from the builder.
If you implicitly trust your neighbour then deal directly with them, but if you have any doubts get your insurance involved regardless of what your neighbour wants. If it goes wrong in a few weeks time, your insurance will kick up a fuss about not reporting an accident at the time!
Had something similar about 15 years ago when someone backed into my Motorbike doing £1000 of damage. He didnt want to involve insurance companies. I got some quotes for him, and when I aproached him about 1 week later with them he denied all knowledge of the accident! Insurance gave me a roasting, but I still got paid from third parties insurance eventually! Could have been worse if he didnt have insurance at all, or given me a false address!