I exclusively used fabric paint, which gave it a stiff, vaguely papier-mache-like texture. I think that works well with this design (it's like hardened frosting!), but I might try out fabric dye or something along those lines if I ever do another one.

Anyway, if anyone out there is thinking of fabric painting a colorpy, be warned that the first coat will probably look rough:

It's possible that using a primer would help, but I can't personally speak to that since I just winged it for this attempt :)

Also, I was surprised how big of a difference the red line on the beak makes in terms of how 'finished' it looked: