The OP has my sympathy. My Mother lives in a block of flats where absurd planning condtions mean one space per flat (even though some are multi occupied shares) and only four visitor spaces for 24 residences. However, it's still my obligation to avoid the selfishly parked cars on the forecourt.

You have two choices:

(a) Pay £650

(b) Refer claim to your insurer and let them deal.

Either will impact on your premium if not your NCB.

I can see the other point of view. Getting three quotes sounds easy, but it's not like car car insurance where you can get a dozen online quotes simultaneously. Body shops are not that common and you have to take your car to each one and wait for them to assess the damage and (hopefully) get back to you with a written quote.Then you have to go back to the original party, get them to approve the lowest quote, arrange a date for your car to be repaired, make arrangements for how the bill is going to be settled, get transport back from the repair shop etc, then get a lift to go and collect the repaired car...

A similar situation happened to my mother. Her car was scratched by a bus. The bus company wanted to resolve it privately. My mother agreed, but it was a lot of extra hassle for her compared to phoning up the insurance company and having a courtesy car sent to her doorstep.