Here's how I would approach this - https://www.dropbox.com/s/tro6kj5txzeeb ... s.PNG?dl=0
There are a few prerequisites:
You're going to need to manually crop out each area to be colored individually, and import them all and line them up together.
Each "coloring block" and "coloring area" needs to have its own layer with the "Force own texture" layer property set.
The color picker is actually just an illusion, its not really unlimited - you can import as many colors into your color block sprite as you can handle (128 would probably give a suitable illusion that it is "unlimited"). I'm pretty sure the link you sent me doesn't have truly unlimited colors either. You can probably get away with just 64 and no one would be able to tell. Seems tedious to set up that many frames, but you only have to do it one time. (Making 12 took me about one minute).