SOL can get reset, but payment is the thing that usually resets SOL. Denial of a claim, submittal of a claim, I would not think impacts SOL. But, SOL is a deep, deep subject.
Definitely sounds like the hospital incorrectly billed this as an auto accident, which it is not, and they simply need to give insurance the requested info to get paid.
If you call the hospital, I would avoid any payment issues or questions, and just tell 'em that the insurance company needs records so that the insurance company can pay them. If you can get the insurance company to send you something, then fax that to the hospital, then you could avoid any direct questioning.
There's enough complexity in this that I'm tempting to say give a consumer lawyer a call. Try either naca.net or myfaircredit.com
You really want a lawyer with insurance experience, not so much one with credit/debt experience. It's really an insurance issue, and if done correctly I think you can get insurance to pay and the hospital to delete it's collection.
Oh, and I would DV the CA. I don't see any liability on your part for this debt, particularly given that the hospital won't give insurance the requested info.
Only thing I can figure is the hospital is worried they'll get sued by you, for whatever reason.