How to remove a tile with floor heating mat
I have repaired the issue I had with this tile. I did not take the advice given above and looked around for an alternative as the risk of damaging the tile-warmer element was too high (and expensive).
I spent a few hours and carefully dug out the grout around the tile and surrounding tiles - this was very tedious as I didn't want to damage the heating element. I vacuumed and cleaned thoroughly. I then mixed up some epoxy grout and with a putty knife pushed the grout as far as possible into the gaps. Since I had used grooved trowel to lay the tiles, the epoxy grout filled under the tile into the gaps. After 2 days I tested the tile and it does not move at all. The epoxy grout is INCREDIBLY strong and hopefully stabilized the tile. I am glad I didn't resort to breaking up the tiles.