Floyd-Steinberg dithering of frames?

deleted . Blender does not support processing an image with a dither filter itself.

You could try rendering to image sequence, then converting it to animated GIF via ImageMagick. I’ve chosen GIF because all *PEG codecs loathe dithering (unless you use a very high quality, and even then).
I also tried to find about post processing in Python (F-S dithering is not really hard to do, really), but couldn’t find any info on the subject.

Adding some info I forgot, in the end I need BMP image sequences for every color channel, not a video file, if that helps finding other options.

That makes the job even easier. You can posterize the images to 2 level per component, then let IM use the dithering it wants, or play with the dithering options.

Seriously, ImageMagick is magic (pun oh, so intended) to operate over image sequences. It’s command-line, though, but versatile as h-!

So just to be clear, Blender itself doesn’t have a dithering filter or dithering option in the renderer, right?

I’m not sure if this helps, but there is an option under the render tab under ‘post processing’ that has a slide bar option for dithering.

That checkbox is to dither from float to whatever precision is selected. Blender (AFAIK) can’t do palletized images, you’ll have to convert them outside.