Ok don't shoot me I haven't read every single update. Having worked in the insurance industry for + 10yrs (ok not doing so now) and also involved in a similar rear end shunt in dec 2013, my advice would be to claim off your own insurance and get them to recover costs from third party. This would also cover any hire car costs and if you have motor legal protection, they will claim back your excess as well. I know it's not an easy fix but in the long run, it will get your car back on the road ASAP and leave them to fight over costs.
I'm insured with direct line and they are highly recommended in my book. As I had a mini 4x4 in dec the hire car co could not give me an exact like for like replacement so I ended up with a free upgrade to a merc e class for 4 weeks! As soon as other insurers admitted liability - which was pretty darned fast, direct line claims dep reimbursed me my excess. All in all had accident 16th dec and received cheque around 9th jan there or there abouts.
Your own insurers would not have dragged their heals as long as you submitted 2 estimates for repair and they would have gone with the cheapest repair bill......they have to be seen to take the more cost effective approach.
Just my opinion having worked in the industry and my similar experience indecember. Good luck hope all goes well for you