Sorry to say it but I think you may struggle not to damage the underfloor heating. I have never heard of it being done successfully with the mesh type. I managed it once but the heating element was the thick wire and not the mesh type, even so I damaged one wire and had to repair it. I would say your best bet is to learn to live with the tiles or budget for new underfloor heating.
Ditto with Kitcheguy. If you do go for it, i would try and break the tile into small bits & go for the areas that sound hollow when tapping on the face of the tile. Once you have got a bit of tile up, try getting somthing under the tile to stop you hammering down on the cable.
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