How can one improve the robustness of adaptive beamformers to signal mismatches?
One approach that I'm aware of is to "load" the diagonal of $\widehat{\mathbf{R}}$ by some value $\sigma_d^2$ in order to stabilize the covariance matrix and make the adaptive vectors more robust to signal mismatches. This is often done in matched field processing and is referred to as diagonal loading or white noise constraint beamforming. 

In other words, the robustified adaptive weight vectors are calculated as

$$\widehat{\mathbf{w}}_\theta=\frac{\left(\widehat{\mathbf{R}}+\sigma_d^2\mathbf{I}\right)^{-1} \mathbf{a}_\theta}{\mathbf{a}^H_\theta \left(\widehat{\mathbf{R}}+\sigma_d^2\mathbf{I}\right)^{-1} \mathbf{a}_\theta}$$

and the output beam power is now computed as

$$\widehat{B}_{mvdr}(\theta)=\widehat{\mathbf{w}}_\theta^H \left(\widehat{\mathbf{R}}+\sigma_d^2\mathbf{I}\right)\widehat{\mathbf{w}}_\theta$$

Here's the result comparing the no mismatch, mismatch due to perturbed array and diagonally loaded cases for the MVDR beamformer for the situation in the question. There is an improvement in the signal power estimation, at the expense of increased background noise power.