Despite strong medical benchmark accuracy, LLMs can exhibit severe multi-turn sycophancy in clinical dialogue, abandoning initial correct diagnosis under escalating pressure. We propose Med-Stress, a targeted stress test framework that evaluates belief stability under escalating pressure. Across nine frontier large language models (LLMs), we find a clear dissociation between medical knowledge and robustness: high initial diagnostic capability does not imply high belief stability, yielding large knowledge-robustness gaps for several LLMs. To mitigate this failure mode, we propose a lightweight inference-time defense, RBED (Role-Based Epistemic Defense), and R-FT (Resilience-oriented Fine-Tuning), a training-time approach that internalizes evidence-based resistance to pressure. Experiments show that R-FT nearly eliminates belief change and substantially improves robustness.