Accurate disease classification from radiology reports is essential for many applications. While supervised fine-tuning (SFT) of lightweight LLMs improves accuracy, it can degrade reasoning. We propose a two-stage approach: SFT on disease labels followed by Group Relative Policy Optimization (GRPO) to refine predictions by optimizing accuracy and format without reasoning supervision. Across three radiologist-annotated datasets, SFT outperformed baselines and GRPO further improved classification and enhanced reasoning recall and comprehensiveness.