If you are dealing with the other driver's insurer then their liability to you is no different than the other driver's personally. If he was at fault, his obligation is to compensate you for the loss you incur, and that would be calculated as the lesser of either the value of the vehicle or the cost of repair, less any value that can be recovered by scrapping the vehicle (though in practice the insurer may deal with this aspect by effectively purchasing the damaged vehicle from you.
So essentially you cannot simply insist they pay for repairs that are not viable, and it will be up to you, once you receive a compensation payment for the car to decide whether you then scrap the car or pay for it to be repaired.
If you can establish the car has value as a vintage car or has been modified for a special purpose which you have or were about to exploit, then you may be able to claim the cost of either modifying a replacement vehicle or, if that would bring the price of a replacement vehicle above the cost of repairs, use this to justify why they should be obliged to repair this particular vehicle, however, be aware that courts and insurers are suspicious of claims that a particular car has special features that make simply replacing it inadequate compensation, so you will need compelling evidence that you used or were about to use any such feature.
If you have your own insurer involved, they will only be obliged to pay what is owed under the terms of the policy, which is usually capped at the agreed write off value which you would have set when you took out the policy.
If you have already entered into a preliminary agreement either with their insurer or yours then you would presumably be bound by it unless there was something improper about the circumstances in which you entered into the agreement.
I trust the above assists.
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