I think you'll be OK. When I used accident exchange there was an insurance policy taken out against AE not being able to recover the costs from the 3rd party insurers. If they failed to get the money then this additional cover would settle the bill.
You may need to attend court to state that you absolutely had to have a hire car, and that you had not other car at your disposal that you could have used rather than hiring one.
You may also need to verify other facts rather than explain decisions regarding the hire car.