The other person's car insurance company has NO duty of good faith with YOU.
You need to consider that it's quite possible that their policy was cancelled, that they are NOT an insured driver under the policy, or that they do NOT agree with you, about the liability of the accident.
They don't have to pay you ANYTHING, until after a judge has found their driver liable, AND found that the driver is covered by the insurance policy. IF there's coverage, AND the driver is liable, they're only "on the hook" for the time period your car is actually being repaired, or a reasonable period to purchase a new car if your car was totalled. And that, is usually $25 a day, times the number of repair days, or a week or two.
Even if that driver IS liable, the insurance doesn't pay out, if the policy was cancelled, or the driver was excluded, or some other policy condition was violated (like they were using it for business use).
Pretend they have no insurance, and procede accordingly. File the claim with YOUR insurance. Or sue the other driver.