Mega Shocked (./14):
I have pretty lame method but I use PNGquant and run my image through it several times.....there can be alot of trial and error involved depending on how obsessed you get.

https://pngquant.org/

DATimage can also reduce colours "Change -mic256 to -mic128 , mic64 etc....

DATimage pathWhereFilesAre\*.png -mtc15 -mic256

I just tried this way of color reduction for a most of time, and tried to reduce the mtc maximally for economy of palettes allotted for characters portraits. But more often the result of direct reduction through DATimage (-mtcX or -mtcX & -micX) looks much less visually neat than a result of reduction to 8bpp through IrfanView and PNGquant, despite of larger nunber of colours.

Here is a -mtc15 -mic256 version of previous picture.

Fichier joint :