If you have collision and comprehensive insurance your insurance carrier has different obligations to you than the person who was at fault for the loss. Your carrier has a contract with you to make you whole and represent your mutual financial interest. You can dispute the value with the at fault driver’s carrier, but your carrier is beholden to the language in your policy including the appraisal clause.

If you make a claim with your carrier even if you aren’t at fault, you carrier will pay for the damage in the moment to make you whole, then demand payment from the other driver’s policy via an internal process called subrogation. In short this means, just because you make a claim with your carrier, doesn’t mean the money for the claim is being paid by your policy ultimately.