Strictly speaking, this is incorrect. Unfortunately, unsuspecting motorists being chased by accident management companies for the cost of the hire car isn't a new phenomenon. See for example:
I am not suggesting that that this will happen to you, however, if you signed an agreement with Auxillis, then it will likely say that you are liable for the cost of the rental car in the event that the other-party's insurer (BT, in this case) refuses to reimburse them.
Again, it's not a common occurrence and I am not trying to get you unduly woried, but for anyone reading your post, its important to know that its incorrect.