I don't know that it's possible to remove the marble tile without breaking it. Marble is full of fishers, and the tile breaks quite easily along these lines if it's put under stress. It will also depend greatly on if the tile was put up with thinset or mortar. Most like it was thinset, so that would be good, because thinset can be disolved with water over time.
What I would do is cut between the tiles (along the grout line) and right through the backer board. I have seen plumbers do this with hacksaw blades (hand-held) and slipped between the tiles (tape the end of the blade you're holding and wear stout gloves or you will tear your hands to smithereens) Remove the marble tile along with the backer board and then soak the whole mess in a bucket with soapy water (this may take several days). Eventually the thinset will soften enough to separate the tile from the backerboard, and then scrub the rest of the thinset off the back of the tile. Let dry thoroughly and then reapply the tile to the newly installed backerboard.
This whole project could be a real bearcat, so you might wish to practice deep breathing and counting to ten! All my best wishes for you on this one.