What I have tried to do is to separate the exponential of the unitary transformation operator to the interaction picture into three different Hilbert "subspaces" like:
$$e^{i\frac{H_0}{\hbar}t}=e^{i\omega_m \hat{b}^+\hat{b}}\otimes e^{-i\hbar\nu|1><1|} \otimes e^{-i\frac{g}{\omega_m}|e><e|t}$$...