Sure, but at any given moment, they can use only ~12 of them (+ the rainbow thing, now that I played a bit and know how it works). Meaning, you can keep 12 pre-filled variants of each mask, one for each color on the main palette, and recolor all masks of a given color when the user picks a new one from the top-left color picker. I bet you can do this update faster than they can dismiss the color picker and tap on the canvas.
I see you have drawing tools there as well, but they seem to be ignored by flood filling (e.g. if I draw a closed circular border myself and use flood-fill on the empty centre, the fill will just paint over my circle border and continue filling to the boundaries of the initial region of the original image) - so they don't even make a counterpoint I was worrying they would.
