Tapes rely on contact to make the mechanical bond, the more uneven/unpolished the surface the less mechanical bond there will be.
We used to use M77 to mount masks to blast granite stones and memorial markers. Then the cleanup would be with Naptha, cause that stuff didn't want to come off the stone.
Chances are high a good rubber cement for countertops/formica installation would work fine, you just need to bray it onto the paddle and the stone evenly, so you don't get any humps, or off angles, when pressed together.
After reading posts on this thread to date, it sounds like the issue is either intrinsic to the tape/plastic or due to water exposure. If the latter, do you use a specific type of Elmer's? Unless they've reformulated the classic school glue, I'm pretty sure it's water soluble...