CBM is not so much a theoretical framework but rather a label attached to a collection of methods and biases. In a way, this reflects the state of research on heuristics and biases, at least in behavioral economics. As many famed findings in the field have come from attempts to "disprove" traditional economic theory, the resulting effects (biases) have often remained without a satisfying explanation in the form of a plausible cognitive process. This has hindered
the formulation of a common theoretical framework in which various biases
could be classified (according to, e.g., underlying cognitive mechanisms, preconditions w.r.t. context and subject, effective interventions).
(To name an exception, a framework covering comparison selection was recently proposed by Simonson, Bettman, Kramer & Payne 2012. The authors cover a number of effects which may commonly be labeled "biases". In addition, various frameworks based on some form of dual-process theory have been proposed, but there is a lot of redundancy between them.)
In addition, the "heuristics and biases" literature has remained largely separate from the psychology of behavior change and, arguably, from cognitive psychology. Look no further than the popular "Nudge" by Richard Thaler and Cass Sunstein: almost all of their implications are derived directly from the biases "as effects" rather than being derived within a more general framework grounded in cognitive process theory.
In short, to the best of my knowledge, the framework that you seek does not exist. I expect that you will find it necessary to devise behavior change strategies one bias at a time.
As a final remark, the heuristics and biases literature has been criticized for tarnishing what may actually be highly adaptative and efficient cognitive strategies (see the illuminating works by Gerd Gigerenzer). Accordingly, many a bias may be exploitable only under rather specific and unusual circumstances.