I was thinking about posting an answer to the question at some point, basically saying that it turns out to be a (fairly) well-studied problem in disguise, and linking to the relevant OEIS pages. In addition, I'd post the numerical things I found for n = 4 (orbit and stabilizer sizes), as well as a picture of a representative colored total from each orbit.

Have you continued / do you plan to continue thinking about the problem from the algebraic side of the fence?