The thinnest material I could find is 1/16" basswood, they (Hobby Lobby) also had 1/16" balsa but it is a poorer grade.
I screwed up, I oriented the pattern 90° from what I should have. The wood did bend as I wanted, I'll see (and post pictures) after letting it dry for a few days. I use a non contact thermometer to determine dryness, check the temperature of the material and then the surrounding area, when the temperature matches, it's dry.
In the November 1998 issue of Creative Woodworks and Crafts magazine, there is an article on fan out birds. The article describes how to bend wood much more than what I am trying. I'm using the article as a guide for how long to soak the wood prior to bending. So far so good.