We know that the insurance company for the person who caused the accident will not pay the bills until the case is totally settled.
You cannot afford to have unpaid bills piled up, especially if you have health insurance coverage. Some health care providers will tell you that you can't submit the bills to your own health insurance company "because it was an accident."
This is wrong but I'll tell you why they do it: If they "participate" with your health insurance company they will only be paid 60% or so of their bill. If they can get you to pay cash or even to wait to the end of the case to get paid out of the settlement they get 100%.
Sound unfair? It's not. You paid a premium for your health insurance. Part of what you pay for is a discount on the "list price" of your doctor's bill. Your doctor voluntarily became a participating provider and agreed in advance to a discount. The doctor gets more patients and you get your bills paid.
You should also submit your bills to your automobile MedPay. Again, you already paid a premium for this. Nothing illegal or immoral about it.
All of this will lead to more money in your pocket later.