The design process went through many iterations and down various dead ends that didn't work for structural or aesthetic reasons. Eventually, I decided to just 3D print the framework. Making the file for printing was pretty easy: I generated an output file using TubeASP, hid all the atoms I didn't need in Mercury, and exported it for 3D printing. I cleaned up the file using Fusion360. All steps are laid out in much more detail in my instructable "How to make accurate 3D molecular models".

However... when I printed it, glued it up and assembled it, I realized that the gaps between frame and acrylic panels were aesthetically, well, revolting. Most people didn't notice it, but it bugged me a LOT. The final design was basically the same, but with hexagonal punches taken out of the back of each cell to hold the acrylic panels. There are some slight irregularities with the wall thicknesses, but I liked it much better than the previous design and it was much stronger and easier to assemble. Thanks to the talented Scott Daigle (current innovator-in-residence @Pier9) for helping me out on the CAD when I got stuck. The STL file for the updated model is attached.