Mine is stuck also. Corroded batteries inside leaked acid at cover and can't get cover to unscrew.
For my blue tooth mac keyboard my last resort was to hack saw a slot through the center of the battery cover down to the battery surface. This split the cover into 2 pieces which I dislodged with a !$$&*-punch and removed with needle nose pliers. The battery compartment thread corrosion was cleaned with a dental pick and the saw cuts were smoothed with a small file. A new cover from Amazon restored the keyboard.