I've glued birch bark onto wood bows using TBIII and it worked well. I don't know anything about how it adheres to glass ... but if macbow says it adheres to glass also, then I think you'd be ok with TBIII.
A word of caution ... any TBIII on the birchbark on the outside surface will cause discoloration when you add a finish so keep the glue on one side only .... don;t be a slob like me.