You have damaged someone else's property, you are liable.Your lack of knowledge of vehicle repairs does not put you in a situation of being able to advise whether £600 plus VAT is expensive or not. That sum of money can be very easily arrived at. The other party has every right to have their car repaired where ever they wish and does not need to have another quotation done. If you do not have the funds available to pay, I would suggest you put it into the hands of your insurer, contact them and advise that you may wish to "purchase the claim from them" and have the claim wiped for your records if you can afford it. You pay your premiums for this exact reason, they are there to cover the cost of any third party claim against you.
Contact the other party and ask how they would like to proceed on a without prejudice basis. Advise that you would like to settle things amicably, but for your own piece of mind, would very much appreciate them getting another quote. The other option is to see if they would agree to have the car inspected by an independent engineer, This would cost you around £200. You then may well find that the £600 plus VAT was cheap. The simplest advice is, it all comes down to what you can afford. If you cannot afford the £600 plus VAT, pass to your insurer.