Personally, I wouldn't clamp it to try to glue it back together. In essence, you are pressuring the handle into a position it clearly does not want to be in (evidenced by the crack). If it were me, I'd clean the crack out, sand it to make it a smooth opening, place a new piece of thin wood inside (like a shim), epoxy it nicely, trim the excess, stain & finish until it matched the original (or, as suggested by other posts here, refinish the entire handle).
I think that if you clamp it down, then release it, it will crack again. It may take a few days/months/years, but it will crack. If you feel it needs to be strengthened, then put in metal dowels & ornamental/medallion heads. Just recess the head into the handle and coat it with a protective coating. It will look intentional, instead of like a repair.
Just my 0.02