I've been through this.
Someone drove into the Gallardo, was a tiny tiny scratch to the rear quarter. The lambo dealership recommended Accident Exchange. My incident was at 5pm, by 9am the next day a Ferrari 430 was on my drive courtesy of AE at a cost of £1100 a day
I had the 430 for 35 days and the total bill was £52k
From the outset I said I would only proceed if I was totally unresponsible for the costs of the hire car. I did have to sign a credit hire agreement for the 430 but AE said that I could take out an insurance policy for £10 that would indemnify me from being pursued for the costs should they not be able to recover the costs from the other parties insurance. I ensured that they sent me written confirmation that I would not be chased for costs.
It was pretty stressul to be fair and took over two years to resolve! I had to send in a load of bank statements etc like they have asked you for. I had the money to pay for the hire car but my arguement was 'why should I pay for a hire car, for a no fault accident? thats what insurance is for!'
They even wanted me to justify why I had a 430 and I just told them that I fully expected a like-for-like car, the gallardo cost me £87k, why should I be without it for 35 days through no fault of my own???
It seems clear that the insurance company have not been able to recover the costs and are now pursuing you for it, do you have copies of anything you signed?