I suspect tuning the laser so it would ablate the outer layer of paint without affecting the inner layer would be extremely difficult to do. The paint is burned off by the laser, so the white underpaint would burn as well.
Using black plastic with white paint might work (but would reverse your color scheme). White lowerctemp plastic (like ABS or PLA) might discolor under the laser-burn as well, so using a white base likely won’t work.
You’ll need a flat surface to use the laser on (it won’t focus consistently on sloped surfaces), so I’d reconsider cutting the panels out of black acrylic on the laser as well.
You could cut into black acrylic and rub white acrylic paint into the etched lettering. You could use 3D printed parts to connect the flat panels. This can be done at odd angles (see this globe I lasercut as an example). It used small pentagonal pyramids in the corners to connect the lasercut panels.
A (somewhat pricy) two-layer acrylic sheet for making nameplates is also available. You laser-etch (or route) through the top layer, exposing the lower layer. This makes for crisp lettering in a contrasting color.
You can get it at Allied Plastics.