In practical customer-support dialogue systems, responses must simultaneously deliver factually grounded information and context-appropriate empathy, yet existing single-stage generation models often exhibit specialization bias, favoring one objective at the expense of the other. To address this limitation, we propose a dual-stage generation framework that explicitly decouples factual grounding from empathetic modulation. Our primary configuration follows a fact-to-empathy order, in which the system first generates a fact-centric draft via structured query interpretation and optional retrieval-augmented generation, then applies empathy-aware tuning conditioned on inferred emotion type, intensity, and empathy necessity. To enable deployment in resource-constrained environments, only the query interpretation module is explicitly trained using knowledge distillation, allowing the overall system to operate with compact 4B–8B backbone language models. Furthermore, we construct a customer-support dialogue dataset designed to reflect realistic interactions involving both informational and emotional demands. Extensive experiments with compact models show that the proposed approach generally improves key dimensions of empathetic response quality while maintaining overall factual performance, thereby helping mitigate the representational entanglement empirically observed in single-stage baselines. Both quantitative metrics and scenario-based analyses confirm that decoupled generation enables a more balanced integration of factuality and empathy than single-stage generation. These results suggest that dual-stage generation provides a practical and extensible foundation for deployable, real-world customer-support dialogue systems.