Harmful memes convey offensive intent through implicit associations between visual symbols and text, requiring a broad understanding of cultural stereotypes and visual metaphors. Small-scale Multimodal Large Language Models (MLLMs) often lack the knowledge required to identify such implicit hate, whereas Large-scale MLLMs, despite their broader knowledge, exhibit systematic labeling bias. To address these challenges, we propose DR-HM, a Distill-then-Reinforce training framework with cognition-aware data synthesis for harmful meme detection, which aims to transfer knowledge from closed-source models while mitigating their biases. DR-HM introduces a six-step structured data synthesis scheme with self-refinement that decomposes meme analysis into a progressive, human-inspired reasoning process from entity recognition to harmfulness judgment. Based on the synthesized reasoning data, we further adopt a Distill-then-Reinforce training strategy. This approach combines a two-stage Supervised Fine-Tuning (SFT) with an Adaptive Group Relative Policy Optimization (A-GRPO) algorithm, which incorporates class-ratio-aware reward weighting and dynamic KL coefficients. Experiments on three benchmark datasets show that the proposed approach consistently outperforms existing methods and achieves an accuracy of 84.7% on the FHM dataset, approaching the reported performance of human annotators.