Glad you like the model, it was created to study the resolution and color change, even if from a more visual than technical perspective. Sadly no sensor to measure the result, so no formula computed; it ended being more a live reference of what to expect.
Also it seems the results change with the print orientation and the coating (maybe even printer settings we have no control over, too), so it would be hard to get a "strong" formula, and it would be best if the maker and operators just provided color profiles. This 3D print system is never going to provide photo quality without some big improvements.
8 bit color is probably converted to 24 bit. And then that has to be converted to CMYK for the print head. But there is no info about the conversions, how are they done, with what precision, etc. The Quick Reference model includes single color gradients in some of the edges (RGB & "CMYK" as per the RGB source file). Sadly Shapeways' 3D viewer renders as a grey object, so you can't see the full layout. They are in the top and bottom edges shown in the photos. The parts were two or more blocks of color meet are not exactly sharp. You can also see there is bleed to the sides (where big areas of squares meet the thing edges with text and bars).
The text is black over white with some minimal edge pixels in grey (in the file) but appears as very dark grey over dirty white (in the print). It all has some kind of extra fuzzyness or halo too, more than font rendering has. The pattern used to test DPI are the bar blocks in the edges, they have black vs white as most extreme case and then different greys. The maximum resolution is achieved with the B&W (0x00 & 0xFF) and the next grey pair (0x24 & 0xDA).
It's a printer problem, in general. You can find 2D printers with resolutions of 600-2400DPI and even more. That just defines the size of the dots of ink. Source material is fine with 300-350DPI (that will go to high quality printing machines for magazines and books) as that DPI is about color pixels (PPI would be more correct), and each "dot" can range from zero to full intensity in multiple steps (256 is pretty common, 8 bit per channel). For old inkjet printers (that's what the 3D printer is based around, 90s tech) 75-100 should be fine. If you want, give yourself some extra margin, say 150-200, but anything more is a waste.