In this work, we treat the problem of multi-task submodular optimization from the perspective of local distributional robustness within the neighborhood of a reference distribution which assigns an importance score to each task. We initially propose to introduce a relative-entropy regularization term to the standard multi-task objective. We then demonstrate through duality that this novel formulation itself is equivalent to the maximization of a monotone increasing function composed with a submodular function, which may be efficiently carried out through standard greedy selection methods. This approach bridges the existing gap in the optimization of performance-robustness trade-offs in multi-task subset selection. To numerically validate our theoretical results, we test the proposed method in two different settings, one on the selection of satellites in low Earth orbit constellations in the context of a sensor selection problem involving weak-submodular functions, and the other on an image summarization task using neural networks involving submodular functions. Our method is compared with two other algorithms focused on optimizing the performance of the worst-case task, and on directly optimizing the performance on the reference distribution itself. We conclude that our novel formulation produces a solution that is locally distributional robust, and computationally inexpensive.