ooh i like your style!
sadly, you can't have partial-transparency in a gif. the format itself does not support that.
as far as i know, no animated image format supports partial-transparency. there are apngs, but they're a very weird format that a lot of devices don't properly support, and i wouldn't recommend using them.
(EDIT: re-reading your post, i realise you weren't asking for a file format that supports both animation and partial-transparency, but software. presumably for the benefit of either pngs or gifs as needed.)
i had no idea firealpaca can't save gifs.
the software i use for animating gifs is an old version of photoshop elements. (i say old, i don't think they still make elements so probably all versions of it are old now). i wouldn't recommend it for animating anything with more than a dozen or so frames, scrolling through the layers is a pain and it will refuse to save a gif over a certain very small size, even though other software can save gifs ten times that size.
i know there's a lot of dedicated pixel-art software out there (i hear aesprite is popular) so you might want to look through those since they'd presumably be less bloated than photoshop for your purposes, and may be better about saving "big" gifs.