I also had a fender bender, not my fault. All states regulate insurance companies, and my state allows the person being made whole to choose the repair shop and parts. The insurance companies try to steer you to their favorite crap parts and cheap fix shops, but you have rights. I’ll bet your state has a legal section in the attorney general’s office that would be happy to answer your questions. Also, many personal injury attorney groups do not charge you for an initial consultation. Many are very happy to give you a list of your rights, including your rights to have your vehicle repaired.
In my case, the front bumper needed replacement. I found out the cost of a custom solid metal bumper was cheaper than the factory one. The insurance company was on the hook for $$ so saving them to $ made sense. The fix was allowed. Do not let the insurance company drive your bus.
Lastly, someone on this thread was very correct. Your vehicle may loose value due to the damage. Other damage; broken shackles, bolts, alignment problems,,, the list goes on may arise later or not be noticed for some miles. True, a little backup fender bender. But you have rights.