Paul, if you don’t have an oscillating multi tool, like a Fein Multimaster...
They make excellent grout and tile removal heads for it. They remove tile with surgical precision, I have done it on 45+ year old bathroom tile and preserved the tiles.
If you find yourself simply rebedding tiles to correct the level under your stand, I suggest this route. It sure beats ripping the whole floor up and replacing it. These heads grind thinset away and grout.
Alternately, a strip of closed cell foam under the entire bearing surface of the stand is all I can suggest to properly fill in voids. The other materials including the neoprene you mention might just slip out of place later. Neoprene is too rubbery. I hope you can figure the leveling out easily. Self leveling concrete could be poured on the floor and tiles laid over it, to make it smooth and level.
Just throwing my ideas out there. I hope it helps some.