Got in on this one a little late...
Other than the 60+ layers you mentioned, you did not indicate that this was a production situation; or what volume of these displays are required. That said, if you're going to continue, or must
continue, cutting by hand, make a cutting template* out out of masonite, or aluminum sheet. It's much easier to mindlessly run your knife around a template than have to concentrate on following a line ... sixty times
* If you have templates other, less skilled, helpers can be enlisted to help with the cutting chore. Watch your fingers while engaging in "mindless" operations...
I'd use a "regular" utility knife (stronger and easier on the hand) rather than a snap off (the blades flex more laterally).
Adhesive? 3M77, or 3M99 - not cheap, but definitely fast.