A Car Rear-ended Me But The Opposing Party's Insurance Isn't Claiming Responsibility (california)

I was rear-ended immediately after successfully merging right into a lane going into an interchange. The driver behind me drove around me to cut into traffic ahead of us, hitting me in my rear driver side with his front passenger side. He continued to try to cut into traffic ahead of us, perhaps to escape, which I was able to take Live Photos of with his license plate and movements. I followed him and ultimately caught up to him to exchange our information (his license, registration, and DL, and my DL - he didn't need my insurance). He apologized repeatedly and I was careful to not respond affirmatively/say "it's ok," but I wished him a good day.

Now his insurance is assigning fault to me. I assume that he made such a claim because I merged in front of him, but if I had done so recklessly, the impact would have been on his front drivers' side and my rear passenger side. Instead, he hit me while my car was already stopped by trying to go around me and cut ahead of traffic.

I have now incurred out of pocket costs from a same-day doctor's visit, 6 sessions of medically-necessary physical therapy, and 1.5 days worth of lost wages. My insurance company just informed me that they are unable to do anything if the other driver's insurance company is not admitting fault, in spite of what I believe is quite damning evidence to prove the other party's responsibility.

Should my insurance company be unable to receive payment from the other driver's insurance company to cover my deductible, is there anything else I can do to strengthen my case? Should I pursue legal representation with a personal injury attorney? Will my insurance rate go up since I would be using my own collision coverage instead of the other party's? If I were to sue in court, would I be able to sue for damages, loss of income, medical costs, and physical and emotional distress?

Apologies for possibly attaching the wrong flair to this; I don't know if this falls under personal injury or traffic.

Thank you in advance!