A complete calculation can be found in <A HREF="https://arxiv.org/abs/1703.05758">Tunnel Splitting in Asymmetric Double Well Potentials: An Improved WKB Calculation</A>

The answer for nondegenerate levels (which seems to be the case asked for by the OP, but the formula given in the OP is for degenerate levels) has the form
$$E_2-E_1=\sqrt{\delta E^2+\Delta^2},$$
where $\delta E$ is the difference of the energy levels in the uncoupled wells and $\Delta$ is a quantity that becomes exponentially small when $\hbar\rightarrow 0$.