let the insurance sort it out.
Similar thing happened to me. A woman did a U turn right in front of me without indicating . I hit the side of her car right between the driver and passenger doors. We exchanged details ,but when the claim came in she said I had driven into the back of her car!
My insurance broker said it was not worth fighting and to just accept it. But I wrote to her insurance and told them what really happened.
I told them to examine her car , which clearly showed a repair to the driver's side and not to the rear as she was claiming.( she lived near me and I went to look at her car) 5 days later I got a letter accepting 100% blame from her insurance and a cheque for the value of my car, £500.
I was quite happy as I had only paid £50 for the car and £10 to get it repaired. I drove it another 6 months before selling it as scrap for £25.