The car belongs in LAW to whoever pays for the car insurance wise, very obvious.
The point is at the moment it is YOUR property, you just settle on the basis that you retain "title"(ownership) the insurers will be only too happy, how the hell someone thinks it's a cat c is beyond me.
A write off simply means not worth repairing considering value after repair and the cost of repair, they may value it totally different to you.
A insurer is LEGALLY obliged to allow you title, so long as it is not cat a or b, do they won't be bothered about, it will save them money having it officially processed as scrap so they will be only too happy.
So just tell them to settle on the basis that you retain title, my gf's car has been written off TWICE by the same insurer, been rear ended twice, 1st time we got £600 odd 2nd time £425, and kept car each time, they didn't even charge for the salvage.