I've never had that situation. However, I've been hit by a person with no insurance twice, and hit by a person whose insurance company prides themselves on their determination not to pay any claims except for their own policy holders.
Based on a discussion with a friend who happens to be an attorney who works for a major insurance company (but it was just a friendly discussion, not a business relationship), I've followed the following approach. I hired a private insurance adjuster and then sued my insurance company. I have contract with my insurance company that gives them the obligation to pay regardless of anybody else's insurance or whatever. Also, once my adjuster had provided a professional assessment of the damages, I had my cars repaired at my expense and to my satisfaction then submitted the bill to the insurance company. That worked for me, but I can't say it would work in different states with different insurance companies. BTW, over a couple of brews, the insurance company's adjuster said that I would be better able to make my claims with a private adjuster since his companies policy was to lowball and stonewall.