I'm not familiar with BodyMovin but it looks like it's just an SVG exporter and it requires some hand coding. If that is truly the case you need to keep everything vectors to maintain a small file size SVG. Puppet Pin turns things to pixels. The distortion in your eye is caused by inexperience with puppet pin. You can fix that, but you can't keep it vectors without some additional work.
You have two options.
- Refine your puppet pin until the animation looks ok, pre-compose, auto trace, and then get rid of the pre-comp in the main comp
- Don't use Puppet Pin and just animate the paths used to create the blinking eye.
There is a third option, forget SVG, fix the puppet pin, make sure that you don't have any identical duplicate frames in the animation, export a lossless movie, import the lossless movie to Photoshop, adjust the timing of each frame in the Photoshop timeline, then export an animated GIF from Photoshop.