Large language models (LLMs) are rapidly changing various domains. However, their capabilities in handling conversational breakdowns still require an in-depth exploration. This paper addresses the challenge of detecting and mitigating dialogue breakdowns within LLM-driven conversational systems. While powerful models from OpenAI and Anthropic excel in many dialogue tasks, they can still produce incoherent or contradictory responses, commonly referred to as breakdowns, which undermine user trust. To tackle this, we propose an approach that combines specialized fine-tuning with advanced prompting strategies, including few-shot learning, chain-of-thought reasoning, and analogical prompting. In particular, we fine-tune a small 8B model and demonstrate its robust classification and calibration capabilities in English and Japanese dialogue. We also validate its generalization on the BETOLD dataset, achieving a 7\% accuracy improvement over its base model. Furthermore, we introduce a real-time deployment architecture that selectively escalates suspicious responses to more resource-intensive frontier models only when breakdowns are detected, significantly cutting operational expenses and energy consumption. Experimental results show our method surpasses prior state-of-the-art specialized classifiers while also narrowing performance gaps between smaller open-source models and large proprietary ones. Our approach offers a scalable solution for robust conversational AI in high-impact domains by combining efficiency, interpretability, and reliability.