Oh. If you look at it from the outside, does it appear to have corrosion at the junction? Are you sure you turned the set screw out enough?
I agree that the safer option is ask an auto glass shop, but if forced to DIY, I might try very gently/mildly heating the mirror piece that slides onto the bracket, and better still to have a cardboard shield around it, diverting heat away from the glass. This should cause the metal to expand.
Another thing I might try with household ingredients, is make a gravy out of corn starch and vinegar, and apply that to the junction. I know, it sounds crazy, but the corn starch will thicken it enough to stay put better, and the acetic acid in vinegar will dissolve some corrosion while it's not harmful to glass, though I wouldn't let it run down the windshield onto the dash, would put down a plastic bag and a towel over that to absorb any runoff. To elaborate, about 1oz vinegar to 1/2 tsp corn starch ratio, heat on the stove or microwave until it thickens. If it does not work after sitting on the metal for a few mintues, rinse it back off because once it dries it will turn into something like GLUE. Maybe forget I even mentioned it, too many what-ifs to consider, but don't use naval jelly as the compatibility of its phosphoric acid with glass is far lower.
Also, even if the mirror won't come off the base, you might be able to get the whole thing off using acetone to soften the adhesive, then have better access to get the two pieces apart.
The more conservative answer is just let the auto glass shop take the risk. Also my prior suggestion about acetone... that was if there was only the metal base plate and the glass to get apart. I would not expose any plastic base plate cover, etc to acetone as I don't know if they are compatible.
