The emergence of autonomous Large Language Model (LLM) agents has created a new ecosystem of strategic, agent-to-agent interactions. However, a critical challenge remains unaddressed: in high-stakes, emotion-sensitive domains like debt collection, LLM agents pre-trained on human dialogue are vulnerable to exploitation by adversarial counterparts who simulate negative emotions to derail negotiations. To fill this gap, we first contribute a novel dataset of simulated debt recovery scenarios and a multi-agent simulation framework. Within this framework, we introduce EmoDebt, an LLM agent architected for robust performance. Its core innovation is a Bayesian-optimized emotional intelligence engine that reframes a model's ability to express emotion in negotiation as a sequential decision-making problem. Through online learning, this engine continuously tunes EmoDebt's emotional transition policies, discovering optimal counter-strategies against specific debtor tactics. Extensive experiments on our proposed benchmark demonstrate that EmoDebt achieves significant strategic robustness, substantially outperforming non-adaptive and emotion-agnostic baselines across key performance metrics, including success rate and operational efficiency. By introducing both a critical benchmark and a robustly adaptive agent, this work establishes a new foundation for deploying strategically robust LLM agents in adversarial, emotion-sensitive debt interactions. The code is available at \textcolor{blue}{https://github.com/Yunbo-max/EmoDebt}.