In Melee Pichus green alt gives him a bagpack, which "objectively" is the best alt in that game. Even though it does not actually reference the Mystery Dungeon series it is reminiscient of the bag used in those games, and seeing as the PMD series holds a dear place in my gamer heart I would love too see it return to SSF2 instead of the overrated (but still good) blue goggles.

Oh.
PMD stands for Pokemon Mystery Dungeon.
I saw the alt, and it actually is great.

Yes, you would need to do this process for each frame, but at least that's only once per character, not once per character per costume.
Another possibility is applying color transformations only to certain colors, like blue for Marth (see sig, the skin tone is unchanged) and pink for Peach. I don't think this is easy at runtime for Flash with ColorMatrixFilter, but it wouldn't be too hard with a photoshop/gimp filter (or maybe shaders, which I know nothing about). Of course, this helps the manpower problem, but not the download size, to which there is another potential solution: Include only the base sprites in the download, and generate palette swaps clientside on first loading the game. Not sure how feasible that is, but it's worth looking into.

There's one last problem of characters sharing the same color in their clothes and body. Fox (white) is the only one to my knowledge. And Marth's hair.

I personally prefer the Red Earband costume on Pichu, it's cute
But the Goggles and Backpack are also great
I'd be fine with any of 'em

dont forget that every one of them has a scarf somewhat similar to pmd as well