Mapping unstructured Cyber Threat Intelligence (CTI) reports to MITRE ATT\&CK Tactics and Techniques is essential for threat understanding but remains largely manual due to severe class imbalance in technique labels and the inability of Security Operations Centers (SOCs) to share raw CTI data across organizations. This paper proposes \textit{FedAug-CTI}, a federated learning framework that addresses both challenges within a unified design. Each SOC, modeled as a federated client, locally augments its private corpus through a language-model-based synthetic CTI generation pipeline guarded by a quality and diversity filter. The augmented data is used to train a two-level multi-task classifier built on a shared DistilBERT encoder with separate tactic and technique heads, optimized under a Dynamic Task Weighting scheme and refined at inference by ATT\&CK-aware hierarchical post-processing. Experiments on the TRAM-ATT\&CK and CTI-MITRE-ATT\&CK datasets show that synthetic augmentation substantially improves rare-class performance in the centralized setting, lifting technique mAP from $0.323$ to $0.440$ on TRAM-ATT\&CK in the centralized setting. Under federated training, the framework retains operationally usable technique-level Recall@10 across both datasets in IID and non-IID configurations, demonstrating the feasibility of privacy-preserving cross-SOC collaboration for automated TTP extraction.