I am given to believe that the value of $\hbar$ is Planck's constant divided by $2 \pi$. Is this not correct, or is the $\hbar$ here different from that encountered in the context of, say, the momentum operator?
The limit $\hbar \to 0$ is short hand for the semi-classical limit of quantum mechanics. Indeed, $\hbar$ is a constant, and one does not have the freedom to tune it. However, to be more precise usually requires a careful statement of the actual physical question and will typically involve looking at states involving large quantum numbers.