Sounds like it’s caught up in the very Claim Protocols that were intended to smooth these things out. Within those, there are various stages with time limitations & some insurers (particularly the cheaper end of the market) will use them as a form of cash flow management.
The reality is, it’s not just your excess & hire charges that are being claimed, Privilege will be claiming their outlay too. The usual BS is to drag stage 1 out as far as they can then either deny liability altogether or, in cases where it’s cut & dry, a 50/50 settlement. The likely next step is Privilege refusing & demanding 100% which, even if that’s done immediately, gives the third party insurers a few months in which to respond. Probably then 75/25 offer & so on.
The best way to shortcut this is to issue proceedings for which they will generally ask you whether you are willing to attend court – if you’re not, they’d likely lose without video evidence etc if the TP did attend.
The whole process was designed to limit what a no win, no fee solicitor could do & earn but when it’s only insurers involved it can become a proper ball-ache.