Although I haven't done it in years, I would use either kip or Italian bookbinder's calfskin, 1 1/2 - 2 oz. To keep the wallet from springing open, just fold your inner piece inside the outer piece and see how much sticks out and cut that excess off. When you glue it up, glue from the outside toward the middle leaving a hump at the middle bottom, which can be sewn or not, I just always stopped my stitching at either side of the hump. Since you're using very fine, thin leather, the hump is not an issue, and the wallet folds up perfectly. That very thin, fine leather also means that the pockets (credit card slots) don't create too much bulk. As long as your leather has some body (as either kip or calfskin do) there is no need for stiffener. Good luck.
Terence Plunkett
Renaissance Leather