I've used birch bark to waterproof a bow backing. It'll work fine with either hide glue or wood glue. If you use hide glue I would size the bark with very thin glue first.
I've applied it perpendicular to the direction you have it, with the side on the outside of the tree for the outside. I split mine less than paper thin using steam and it remained pliant enough to work dry after. I expect thick bark may crack once it's bent.
I've had good luck and plan on doing it again, I think it's a great look! There are some sweet turkish and egyptian bows that were covered with birch. Since paper birch grows in neither region I would assume it's benefit was well worth it.