AIUI (and not wanting to take away from Tim, who actually does it for a living!), and to take it to the logical conclusion:
Third Party Insurer (TPI) decides car is uneconomical to repair, wants to write it off and pay out. OP doesn't accept this, so throws it back to them. TPI do not budge. Ultimately OP would end up in court trying to convince a judge to get the TPI to make his car perfect again regardless of the cost which is obviously just ludicrous to attempt so tbh you can forget that approach. What the OP would actually need to do is then start the process from his insurer, who would then contact the TPI to sort out.
Now here's the rub: In the policy document, the OP has agreed that his insurer may deal with any claim as they see fit. I can promise you that they will push back against the other insurer, but if repairs are going to be more than about 60% of the value of the car then both companies will agree that a write-off is most sensible way forward. Courts will agree, the ombudsman will agree, so no go. If you don't want them to write it off then fine, see if you can get them to contribute a sum towards the repairs in lieu of payout, but ultimately you may well just have to accept the inevitable. If it was a particularly rare or valuable car then you'd have a better chance, but otherwise reality and common sense has to kick in somewhere.