Federated learning enables multiple medical institutions to train a global model without sharing data, yet feature heterogeneity from diverse scanners or protocols remains a major challenge. Many existing works attempt to address this issue by leveraging model representations (e.g., mean feature vectors) to correct local training; however, they often face two key limitations: 1) Incomplete Contextual Representation Learning: Current approaches primarily focus on final-layer features, overlooking critical multi-level cues and thus diluting essential context for accurate segmentation. 2) Layerwise Style Bias Accumulation: Although utilizing representations can partially align global features, these methods neglect domain-specific biases within intermediate layers, allowing style discrepancies to build up and reduce model robustness. To address these challenges, we propose FedBCS to bridge feature representation gaps via domain-invariant contextual prototypes alignment. Specifically, we introduce a frequency-domain adaptive style recalibration into prototype construction that not only decouples content-style representations but also learns optimal style parameters, enabling more robust domain-invariant prototypes. Furthermore, we design a context-aware dual-level prototype alignment method that extracts domain-invariant prototypes from different layers of both encoder and decoder and fuses them with contextual information for finer-grained representation alignment. Extensive experiments on two public datasets demonstrate that our method exhibits remarkable performance.