Can I install heat mat and backer board over cork flooring?
When I am redoing a bathroom and adding the needed backer board to a subfloor assembly, I would remove all existing finish floors no matter how many to get back down to the original subfloor. Then evaluate that, repair it if needed, add to it if needed to make it stiff enough for tile, then add the one layer of 1/4" backer board, then the heating wires, screed, then thinset and tile.

All these layers together will add up to a reasonably thick floor, if the all original flooring is not removed, you will have a potentially sizable rise in height from one floor to another