So my final plan for printing the panel is as follows:

1. Paint the semi-transmissible acrylic sheet with black paint
2. Laser cut the panel, using the engrave setting to burn through the paint.
3. Laser cut a mask for the engraved areas on paper.
4. Do any additional drilling necessary.
5. Spray paint over the mask in order to paint the cut edges.

This will work, I think, but I want to say that painting acrylic is way more of a PITA than expected. Don't expect to just grab a can of spraypaint and give it a once over: you need to sand it, prime it, sand it again, then let it dry for 24 hours, then spraypaint it. If I were doing this again, I'd think about whether I could make this work using the engraveable two-tone acrylic sheets that inventables carries (maybe with another sheet of something as a diffuser?)