Faaaaairy, faaaaairy, faaaaairy, faaaaairy, faaaaairy, faaaaairy!
I was wondering why this seemed familiar! I forgot about that Pokemon Stadium game, awesome throwback and use of color!
Set background color as transparency color in both your layers and your frames
That Clefairy game in Pokemon Stadium, haha.
I don't use GG sadly, but I did find this video in case you haven't tried that yet - it mentions PNG files but hopefully it works for animations as well.
Also, that's a really cute Clefairy
(Anyone know how to make a gif transparent, btw? I can't seem to do that in GraphicsGale - not entirely sure whether that's a software limitation or I'm just dumb.)