TB III is awfully strong if you use it properly.
Two tips. As mentioned, it does not fill gaps well, so do a dry run clamping your lam and backing together and make sure everything fits perfectly, that you have enough clamps, etc. Also, keep your workspace really clean when gluing, because even a little wood crumb can cause you little problems.
Secondly, I like to leave bows glued with TB III on the form for several days. IT dries pretty well overnight, but does contain water, and might be dry, but not fully dry or something. At leasty I have noticed a little "creep" off the form if I rushed the bow, started bending it for floor tiller, etc. too soon.