Here's a suggestion for you. Remove any shelves in your display case.
Create a carpeted piece of plywood and attach the tiles to this board using
heavy duty VELCRO. It will stick very well, even though the tiles are heavy
to the carpeted surface (indoor outdoor....industrial). The VELCRO can be
removed from the tiles at the end of the showing without harming them and you
can also keep the carpeted board for next year's display. Hope this is a
suggestion that you can use. I did this and it worked great!