You could back it with a thin piece of hickory or something similar and induce reflex upon glue-up, that should kill two birds with one stone... add weight and reduce string follow.
When you reattach the handle piece, be sure to prep the gluing surfaces well. Get all the old glue off and get down to new wood. Whether you finish the gluing surfaces coarse or smooth depends on which glue you decide to use.
For handle pieces, I like to prep with a toothing plane blade and use an epoxy with gap filling properties.