Hi Rich.
Dear heavens above, you really do have the excrement Gods dumping on you at the moment.
I think this is going to depend upon the actual contracts.
As far as you are concerned you have a contract with your insurance company to cover all the claimants costs if you're involved in a collision.
So, I don't think that your insurance company can "refuse" to pay out. They might challenge the amounts if they are excessive but they are still responsible. For instance the claimant can't hire a Merc E class if they run around in a Polo. From what you described above this sounds like what they're contesting.
The bit I'm not sure about is when it's a business insurance so that may well change things.
Ask your insurance company for exact clarification of the issue with paying the care hire. What costs are they challenging - are they looking at £500 or £5000?
It might, of course, just be part of the "game" between insurance companies to get this claim pushed to the top of the pile to be settled more quickly.
Good luck in getting it sorted.