Over at the Pen Blank Press thread, I line the wood clamping faces of the press with a layer of acrylic specifically because the wood glue comes off so easily.
I’ve never found wood-to-acrylic to bond well and always look for ways to incorporate a mechanical bond instead. CA glue is better and epoxy probably best, but none of these is as solid as a mechanical bond. Having put it to use now on several occasions, I can attest to the ease with which I can get a 1/8" disc off of the acrylic without damaging it. Couple taps with a wooden mallet and presto.
For a design like yours my most likely approach would be to cut two strips of wood to glue to the inside of the lid on either side. They would be face-glued to the side of the lid with the edge butted against the acrylic. This provides a super-solid wood glue joint supporting a mechanical fastening that is pretty much guaranteed not to fail.
As to design, having the edge of the support come out as far as the existing edge would add way too much visual weight to the sides. Instead, I’d probably make the strips about 3/4 of the height, 80% of the width, and incorporate the kumiko design so rays fanned out from the top toward the edge. The short sides would slant in along the same angles as the hexagons in the top design. A contrasting wood such as walnut would cement the look as more intentional and artistic than as a compromise in the name of functionality.
Of course the assumption implicit in the phrase “my most likely approach” is that I could design a pattern like that in the first place. Because if I could design that box, the supports I described would actually be possible. In real life my supports would be closer to a couple of popsicle sticks glued inside the lid if I had to design it myself. Big props to you on the design and execution.
Hey, I just realized - sometimes it pays to think inside the box after all!