I suppose it’s really a matter of taking the pay out value up with the at fault owner of the other car. In the end, insured or not, they are at fault and can be sued for the loss/damages.
So, it really just a matter of determining the value of the car. I believe that there are firms that will provide a written valuation for a fee. It’d be handy to have some sort of formal valuation (and the implied backing/support of a professional valuer) to present to the insurance company. If they don’t accept that valuation, then take the owner to court and let the owner take it up with their insurance company.
I think you'll find the dealings will be between the OP and his insurance company...The OP's insurance company will then take it up with who ever from that....
The Op should get values of similar conditions cars currently for sale and use those values to argue whether he should get more that what it was insured for, if that's what the amrket is saying.
The OP taking the at fault driver to court is fraught with danger.....I personally took an at fault driver to caught for damages to a car and whilst I won the case, the person I took to court gave the middle finger to the court and took off interstate....
When I enquired at the court about this. the response was, I had to go to the state the person bolted to and go through the bulldust again....(this is just a short version, but people should get the drift of what goes on)
So at the end of the day, the Op should only deal with their insurance company and his insurance company can go through all the headaches of recooping the money for the car