Automated International Classification of Diseases (ICD) coding is a core medical-coding task for billing, epidemiology, and clinical decision support. Generative large language models (LLMs) are often reported as weak medical coders, but this finding mainly comes from inference-time settings such as prompting, retrieval, reranking, or tool use; the role of task-specific post-training has received limited systematic study. We present a controlled empirical study of post-training for generative ICD coding, comparing discriminative baselines with LLM coders across prompting, supervised fine-tuning (SFT), and reinforcement learning (RL) under a common protocol and metric set. To our knowledge, this is the first comprehensive controlled study of post-training for generative ICD prediction. We further introduce Progressive Hint Injection (PHI), a diagnostic missed-code curriculum within Group Relative Policy Optimization (GRPO) training. Our results show that prompting-only evaluation substantially underestimates the potential of LLMs for ICD coding. SFT provides the main capability jump, GRPO further improves code-set prediction beyond SFT, and PHI provides targeted gains on macro-level performance. These findings suggest that the main bottleneck is not the generative formulation alone, but how the model is adapted and optimized for full-taxonomy recall. We release our code, data splits, and checkpoints at https://anonymous.4open.science/r/LLM4ICD.