You do not indicate the amount of the repairs the insurance company is claiming is required but if the value of the car is correct, repairs must be at least $8000.00. Please note that there is no requirement in NY that you use an insurance company recommended body shop to perform the repairs. If you have a "family" body shop or mechanic, you should go for a second opinion (another estimate) concerning the amount of work needed. If your concerns are legitimate, the only way to "ask" the insurance company to total the vehicle is to have someone else give a competing estimate. Also, you should make sure your idea (and the insurance company's) value of the car is correct by visiting Kelly's or Edmunds online and also checking local classifieds. Lastly, make sure you get a copy of the insurance company's estimate to show to your mechanic. You are entitled to the paperwork.
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No you have no right to ask for it to be totaled. If the value of the repair is under the book value that is what you will get. If you no longer want the vehicle you can cash the check and sell the vehicle - you may actually get more than book value doing it that way.