While large language models (LLMs) have shown promise in healthcare, their application for rare medical conditions is still hindered by scarce and unreliable datasets for fine-tuning. Hyperhidrosis, a disorder causing excessive sweating beyond physiological needs, is one such rare disorder, affecting 2-3% of the population and significantly impacting both physical comfort and psychosocial well-being. To date, no work has tailored LLMs to advance the diagnosis or care of hyperhidrosis. To address this gap, we present LLM4Sweat, an open-source and domain-specific LLM framework for trustworthy and empathetic hyperhidrosis support. The system follows a three-stage pipeline. In the data augmentation stage, a frontier LLM generates medically plausible synthetic vignettes from curated open-source data to create a diverse and balanced question-answer dataset. In the fine-tuning stage, an open-source foundation model is fine-tuned on the dataset to provide diagnosis, personalized treatment recommendations, and empathetic psychological support. In the inference and expert evaluation stage, clinical and psychological specialists assess accuracy, appropriateness, and empathy, with validated responses iteratively enriching the dataset. Experiments show that LLM4Sweat outperforms baselines and delivers the first open-source LLM framework for hyperhidrosis, offering a generalizable approach for other rare diseases with similar data and trustworthiness challenges.