Is the following true if $f$ is a scalar?
$f\,(\nabla\circ\textbf{B})=f\frac{\partial B}{\partial x}+f\frac{\partial B}{\partial y}+f\frac{\partial B}{\partial z}=\frac{\partial B}{\partial x}f+\frac{\partial B}{\partial y}f+\frac{\partial B}{\partial z}f=(\nabla\circ\textbf{B})\,f$
If you are trying to determine the commutator of a scalar function with the divergence operator then what you need to compute is,
$$[f,\nabla \cdot]B = f \nabla \cdot B - \nabla \cdot (fB) $$