Auxillis are not your insurance company and have no relation to them, other than your insurer obtaining a referral fee for passing your details across to them. Your insurer cannot refuse to repair your car as that would be breach of contract. You can advise that you wish to make a claim via your own policy rather than Auxillis. When you enter a credit agreement, you are liable for all cost till they are reclaimed. They principle part of the agreement is that you are "impecunious" i.e. you do not have the funds to hire a car or pay for the repairs of the car yourself. Auxillis also profit from your claim, they force their repairers to repair your car at much lower rates than the market rate, then charge the at fault insurer the "retail rate" you are entitled too.
This is covered under the case law "Coles V Hetherton": www.honestjohn.co.uk/insurance/coles-v-hetherton-w.../ This difference in "monies" is your money not theirs, when you sign that agreement you are agreeing to them having your money.

Any claim made under "Tort Law" is about financial loss under what is called a "Chattel" i.e. your vehicle, not about repairing your car. You may find your vehicle is not repaired to manufacturer's standards or has bits of the repair process omitted as the repairer is forced to repair for as little as possible so the claims management company make more profit. I would ask them to provide evidence it will be repaired to manufacturer's standards and that the vehicle has pre and post diagnostic checks carried out on it whilst being repaired before agreeing to anything.