TB III is awesome stuff, and fine for bows, but the epoxies and resin-based glues fill gaps better. TBIII needs really good mated surfaces and enough, but not too much clamping pressure.
Glueing a longer slat on the back doesn't seem to work, unless you make it a power lam under a backing. I mentioned tied-on handles, though, and I did make a couple bamboo bows with a tied-on stiffener on the front and handle build ups on the belly side.
The bamboo slat was 1-3/4" wide about, and the handle was an inch or a tad more. I boiled and bent a short piece of bamboo, maybe 12" long, to curve forward like a shallow "C". Then I lashed down the middle of that piece on the back and some little chunks on the belly to fill out the grip. Then I clamped the tips of the "C" down to the limb and lashed them in place about where the fades would be. That worked.