SO... I am going to try and recreate the puppet with FXG files.
SO, I put all the body parts in frame one of the Flash timeline, and exported a FXG. I then imported it into Adobe Illustrator. I bypassed the issues I had before with body parts disappearing by making sure everything was broken up and (important) the vectors were colored with a color number, not a bitmap swatch (which I had done before to make it easier to universally change colors in Flash.)
I imported it into After Effects. To recap: