For the base of my tiles I decided to use 1" thick extruded polystyrene. This is stiffer and more solid than the expanded version, and far lighter than MDF or plywood. it sells locally for $15 a sheet which will do 2 tiles with one knife cut.
The carpet is then glued on with No More Nails which doesn't seem to interact with the plastic an any way. The carpet tiles themselves also have residual glue from whatever held them down in their last life, which gives a bit of tack round the edges. I will add some rare earth magnets to help things hold together on the table top.