Accurate brain tumor segmentation from multi-modal magnetic resonance imaging (MRI) is a prerequisite for precise radiotherapy planning and surgical navigation. While recent Transformer-based models such as Swin UNETR have achieved impressive benchmark performance, their clinical utility is often compromised by two critical issues: sensitivity to missing modalities (common in clinical practice) and a lack of confidence calibration. Merely chasing higher Dice scores on idealized data fails to meet the safety requirements of real-world medical deployment. In this work, we propose BMDS-Net, a unified framework that prioritizes clinical robustness and trustworthiness over simple metric maximization. Our contribution is three-fold. First, we construct a robust deterministic backbone by integrating a Zero-Init Multimodal Contextual Fusion (MMCF) module and a Residual-Gated Deep Decoder Supervision (DDS) mechanism, enabling stable feature learning and precise boundary delineation with significantly reduced Hausdorff Distance, even under modality corruption. Second, and most importantly, we introduce a memory-efficient Bayesian fine-tuning strategy that transforms the network into a probabilistic predictor, providing voxel-wise uncertainty maps to highlight potential errors for clinicians. Third, comprehensive experiments on the BraTS 2021 dataset demonstrate that BMDS-Net not only maintains competitive accuracy but, more importantly, exhibits superior stability in missing-modality scenarios where baseline models fail. The source code is publicly available at https://github.com/RyanZhou168/BMDS-Net.