I have puzzle mats as well, and IMO nothing works very well on them. Gliding discs, Valslides and moving discs do move, but not as slickly as on carpet. Fine for some moves, but it may cause too much friction that can be tough on the knees.
I suggest getting a piece of carpet remnant, or a large bathroom rug ro put down when you need to use the paper plates/gliding discs/furniture movers. The bathroom rug is good, because they have a non-slip backing, and they are also more flexible than regular carpet and easy to roll up and tuck away when not needed.