LEGO Enclosure Clasp
I believe I may have found a workable solution and the "best" solution based on the criteria:

I still wasn't able to find the rubber Technic connector part I wanted, so I kept looking at other connectors and how they fit together, taking special care to note how much space they would take up.

I looked more closely into the "Pop-Up Book Clasp" option (in the question), but that seemed like it would require a complete redesign of a majority of the tower (at least the landing area, anyway). The actual latches in use for that require a very specific spacing, and I couldn't achieve that with the existing layout. Not to mention, those changes would make certain "features" virtually impossible to implement as I intended.

Because the landing area is intended to double as a storage area, I wanted to keep it as clutter-free as possible, but I also want a "clean" look so it doesn't look thrown together. The piece(s) I finally came up with to implement the "latch" for the covers are  the 1x2 plate with vertical holder (60470) combined with the 1x2 plate with stick Ø3.18 (48336). I managed to conceal most of it in the design, and it should operate as expected.

I'm still open to suggestions or alternatives, and it may be a little while before I get all the LEGO I need together to actually try this as a physical build but, for now, I believe this will get me where I want to be. If you want to look at the changes, here's the link to the 3D version of my v1.3 dice tower model on MecaBricks, and here's a static screenshot.