I want to play devil's advocate for a moment.
If there was an accident, whether it was your fault or the other person's fault, wouldn't most reasonable people file an insurance claim? .
(I guess it depends on your definition of "reasonable".)
But if YOU damage MY car, you can be damned sure that I will not be filing a claim on MY insurance! I will file a claim against YOU. not your insurance, not my insurance, YOU. You can pay it or submit it to your insurance. I have no obligation to deal with any insurance company. (Other than to say "you owe me this much.")
For some reason people seem to allow insurance companies to be a surrogate for the legal system wherein liability is assigned...so we "file insurance claims" and allow the insurers "policies" to determine the extent of liability... At the end of the day, it is cheaper to settle with insurance, but you don't need to start there...
Oh, there is another BIG reason to not just "file with your insurance": "your" insurance will not pay diminished value- so that's a non-starter.
There is another issue at play here:
There is a potential claim that this wasn't just an "Accident" but that the dealer, by virtue of poor 'key control' policies, or lack of compliance to 'test drive policies' was WILLFULLY NEGLIGENT in this situation. Perhaps there were other claims like this? Hmmm? This could be a ripe situation for an attorney to explore...perhaps, I dont know... but if I was the dealer and there was any history of this, I'd want this issue settled and done before the OP stumbled upon an attorney with some free time to invest in an exploration.
This latter point is something the dealer and their insurer should consider in deciding on a "fair and equitable" resolution. (That phrase being "screw you" in 'business speak')