I was just in this situation however, i was the one to back into someone. I damaged her back bumper and scraped the side of her car. Nothing too bad.
She wanted me to pay out of pocket. We agreed that she would take it to the body shop for a quote. The body shop agreed to send me the repair quote. It was about $500, which is lower than my deductible. The body shop agreed to charge it to my credit card when she brought the car in to get repaired. They provided me the quote and receipt of payment of the work performed.
She also needed a rental for a couple of days. Not one rental company would agree to let me pay with her on the rental agreement. So she sent me the rental agreement and I paid her directly for it.
It took a leap of faith for doing it this way, but in this state you have 10 days to file a police report. She did the quote and car rental with 5 days.