Then there is the possibility that if you turn it in to your insurance company, they may raise your rates even if they subrogate against the other carrier. If they never find out about it, they won’t have that opportunity.
You can tell either company where YOU WILL have it fixed. They can’t demand where you take it.
No fault only applies to the first set dollar amount of YOUR MEDICAL expenses. It’s designed to keep you from having to sue “the other guy” to collect small change. It does not take care of any damage to your car. IF you were injured, you have no choice but to tell your insurer for your no fault coverage to kick in.
As a quick check to see if there is suspension damage, stick three or four fingers between the front of the tire on the damaged side and do the same on the undamaged side. If the same number of fingers fit, the damage to the suspension likely to be light. My guess is it’s bent. If you were pushed into a curb, check BOTH ends of the car.