The entire thing is essentially already a threshold dithering process, pick the closest available color and throw away the error. So one could certainly extend this to a more sophisticated dithering process and maybe even get rid of the first dithering process from 24 bit to 12 bit and do it all in one go. This also made me realize that there is no good reason why the author is iterating through all red, green and blue values, one could just compute the closest ones.

Huh, that's a really good point: wouldn't using a simple error diffusion dither give decent results here? We basically have 16 palette colors, plus three individual channel-delta ones, so all we have to do is check which of our nineteen options has the lowest error, instead of the normal threshold approach.

The thing is, you do not have the palette in advance, that is kind of the main point of the article. As the author says, one could build a palette in advance, but it is not clear that one of the common algorithms for that would yield a really good one as they look at the entire image but in this case we mostly need colors for strong edges that are not handled well by the HAM encoding. So maybe a good solution would be to first only use HAM and record all the errors, the build a palette that reduces those errors as much as possible, and then encode it again with HAM and the palette.

The thing is of course that the errors propagate with HAM as you can always only adjust one channel and so any change to a pixel will or at least might also affect errors of subsequent pixels. On top of that you might have error propagation from a dithering algorithm. But I think here I have reached the limit of my reasoning capability without having to pull out pen and paper and really digging into it. Final note, I would probably try to find a iterative algorithm that tries to reduces the error in several iterations as I think jumping directly to the best solution is probably pretty hard.