While it normally a tough thing to do, although there can be circumstances when the responsible party is amiable to it..and sometimes you can get a 3rd party involved to essentially get you there.
The responsible party is only liable to pay for the cost of repairs up to the value of the vehicle...whether you want it fixed or not. They never have to pay more than the value of the car...that is the most you had to lose.
However, if they pay you the amount of the repairs they have no interest in the car. But if it is totaled the paying party gets to own the wreck. So, if the wreck as a junk value that is more than the difference between the cost of repair and the full value of the car, they may be interested in paying you the (higher) full value and selling the car for the junk value, essentially costing them less. But, they also have to be interested in messing around with the whole thing more, which insurance companies generally aren't - every action and decision costs them more money.
If you can make it easy for them, basically finding someone who will buy it from them, they may do what your asking. But once again, there isn't anything stopping you from doing exactly that...take the $ for the repair and sell the junk to the interested party...adding the 2 together may even get you more than full value!
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