I understand what you want. The smaller the image, the more difficult to animate subtle and slow movement on it, because each pixel is a bigger part of the image. That's when
sub-pixel animation is useful.
This link explains it in great detail. Is a short read, and helps a lot when you want soft movement on small pixel art.
I made a fast attempt at it on the right eye. Is not as subtle as it probably could be with more work on it, and I only added one new color to the palette, so it kinda looks weird when you start zooming in, but I believe it looks decent at 1x and maybe at 2x. Also, the anti-aliasing makes the eye look a little bit more circular too.