Is it a good idea to bring a Quote when an insurance company decides what to pay for a car accident.?

My car was hit by a driver who made a right turn with a red light as I was going straight through a green light. He claimed he did not see me but claimed fault for the accident to me. He said that he would not be filing a claim because there was no damage to his truck. I called his insurance company and reported... show more My car was hit by a driver who made a right turn with a red light as I was going straight through a green light. He claimed he did not see me but claimed fault for the accident to me. He said that he would not be filing a claim because there was no damage to his truck. I called his insurance company and reported the accident then called my insurance company and filed the claim. The rear quarter panel and rear fender will need to be replaced and this is it job that typically cost around 9.5k. I was wondering if it was a good idea to get a quote from the BMW certified collision repair center before I dealt with the insurance company. I was also wondering if it would be a good idea to bring up this quote during the claims process as it is an expensive job and I do not want to be shorted by the insurance company.
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