A complete calculation can be found in Tunnel Splitting in Asymmetric Double Well Potentials: An Improved WKB Calculation
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 $\hbar\rightarrow 0$.