Would it work if I glued a new closet flange to an existing closet flange?
Usually if the flange is not high enough, a larger wax ring is used or stack two wax rings together. Yours sounds like it is lower than any I have ever encountered. A good plumber with an internal pipe cutter could cut it with out damaging the floor, add a coupler, and with some proper material to hold the flange at the right height, reset a new flange. Seal that to the surrounding finished floor, so if the wax ring DOES leak, it will show on the floor, not go down around the pipe.