Accurately predicting click-through rates (CTR) under stringent privacy constraints poses profound challenges, particularly when user-item interactions are sparse and fragmented across domains. Conventional cross-domain CTR (CCTR) methods frequently assume homogeneous feature spaces and rely on centralized data sharing, neglecting complex inter-domain discrepancies and the subtle trade-offs imposed by privacy-preserving protocols. Here, we present Federated Cross-Domain CTR Prediction with Large Language Model Augmentation (FedCCTR-LM), a federated framework engineered to address these limitations by synchronizing data augmentation, representation disentanglement, and adaptive privacy protection. Our approach integrates three core innovations. First, the Privacy-Preserving Augmentation Network (PrivAugNet) employs large language models to enrich user and item representations and expand interaction sequences, mitigating data sparsity and feature incompleteness. Second, the Independent Domain-Specific Transformer with Contrastive Learning (IDST-CL) module disentangles domain-specific and shared user preferences, employing intra-domain representation alignment (IDRA) and crossdomain representation disentanglement (CDRD) to refine the learned embeddings and enhance knowledge transfer across domains. Finally, the Adaptive Local Differential Privacy (AdaLDP) mechanism dynamically calibrates noise injection to achieve an optimal balance between rigorous privacy guarantees and predictive accuracy. Empirical evaluations on four real-world datasets demonstrate that FedCCTR-LM substantially outperforms existing baselines, offering robust, privacy-preserving, and generalizable cross-domain CTR prediction in heterogeneous, federated environments.