I believe that AR coatings are only applied to the inside of the watch crystal. If this is the case, then the scratches you are seeing are not to the coating, but to the sapphire itself. Sapphire is incredibly hard to scratch (bravo for finding a way to scratch it), but that also means it is equally as hard to polish.
You will have to polish this crystal with diamond paste, and a good deal of patience. Otherwise, replacing the crystal may be a better option.
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