Not sure about where you live but up here in Canada you can get your car fixed WHERE YOU WANT. The insurance compnay can say what they want but you decide what gets done since it'* your car and not theirs.
But since it is from the other persons insurance it may be a bit harder but hey, they hit you so maybe if they brought it where you want you wont decide to be as hard on them for the medical claim
Been there done that, I was rear eneded while I stopped for a bus letting off kids. Had I not been there she would have hit the kids, and I got charged, go figure. It was a woman driver that hit me, her woman friend as witness and a woman cop that investigated. Lets just say it got thrown out of court cause I took pics of the accident along with the "no skid marks " behind the car. Plus once I got a hold of a bus driver that seen the accident she was pissed that I got charged cause she though the car was headed for her and her testamony never showed up on the police report, go figure.
I ended up at the chiropractor for a good little while because of that but since I didn't let them call the ambulance to check me out I couldn't do anything.
Good thing about the situation was they replaced my back bumper , realigned the box on the truck and fixed a few dents on it. Her car got written off, that'* how hard she hit me.
Kind of off topic, sorry, but needless to say it'* your car and you should say where it is and isn't gonig to get fixed.