Ensuring resilience to Byzantine clients while maintaining the privacy of the clients' data is a fundamental challenge in federated learning (FL). When the clients' data is homogeneous, suitable countermeasures were studied from an information-theoretic perspective utilizing secure aggregation techniques while ensuring robust aggregation of the clients' gradients. However, the countermeasures used fail when the clients' data is heterogeneous. Suitable pre-processing techniques, such as nearest neighbor mixing, were recently shown to enhance the performance of those countermeasures in the heterogeneous setting. Nevertheless, those pre-processing techniques cannot be applied with the introduced privacy-preserving mechanisms. We propose a multi-stage method encompassing a careful co-design of verifiable secret sharing, secure aggregation, and a tailored symmetric private information retrieval scheme to achieve information-theoretic privacy guarantees and Byzantine resilience under data heterogeneity. We evaluate the effectiveness of our scheme on a variety of attacks and show how it outperforms the previously known techniques. Since the communication overhead of secure aggregation is non-negligible, we investigate the interplay with zero-order estimation methods that reduce the communication cost in state-of-the-art FL tasks and thereby make private aggregation scalable.