digdj1988
I agree with previous comments that in these circumstances there is little point disputing fault. The best you can achieve is 50/50 but this will still be counted as fault by Insurers and reduce NCD. The other driver and their Insurer will fight back as they will not want 50/50. This would mean the other Insurer paying out for damage to their own Insured's car and the other driver having a fault claim which may affect their NCD.
Also your Insurers have already admitted fault and I doubt they will back down. They will pay for the 3rd party damage whether you like it or not and it will affect your NCD. Even if you managed to pursuade your Insurers to stop the claim until 'fault' is reviewed, you will then have the 3rd party, their Insurers, their lawyers and your own Insurers on you back. This could delay matters for months, which your Insurers will not allow to happen as they could be subject to a higher 3rd party claim for additional costs e.g hire car.
As ATM suggests the only option is for you to pay for the damage to your own and 3rd parties car yourself. I think this will cost you more than the loss of NCD over the next few years. Find out the cost of repairs and ask your Insurers for approximate additional Insurance costs that you may incur over the few years, until your NCD is built up again. Your Insurers may not be willing to estimate additional cost, so you may have to roughly calculate this yourself.
I agree with dacouch, that fighting this is futile. Take it on the chin and put it down to experience. If you do fight this and convince everyone that the other driver was at fault, please post back.
Good luck
Huckster