Fender bender – I’m not at fault. Do I use my health insurance to seek medical care?

Never really gotten into an accident but the other guy is at fault. I went to an urgent care and I used my medical and pharmacy insurance. I may have a few follow up visits.

When the adjuster calls, do I ask for the copay I paid? Or will the whole claims be submitted differently?

I read somewhere my health insurance may reject my visits due to motor vehicle accidents so I have to let the whole charge go through their “medical coverage /PIP under car insurance”. But how do I seek care without paying the copay..? Should I call my health insurance?