With the help of your observations, I can confidently say that the different dither methods don't play much of a role after filtering with a better palette has already been done. So palette-filtering -if we can refer to it as such- is the actual MVP in resolving this issue.

animated webp may also be an option

Hehe :P , I'll take note of this and perhaps resort to it the next time. The whole palette-filtering stuff seemed like some occult incantations that somehow worked. But I would much rather use a different (sane) format instead.

Again, I would like to stress that I've very much enjoyed this interaction! While it's been (mostly) totally unrelated to the original post, this has actually been one of the most informative interactions found within its comments. Therefore, thank you!