While large language models (LLMs) have been introduced to knowledge graph completion (KGC) to improve multi-hop reasoning, most existing approaches still rely on embedding models for candidate pruning, which limits their structured reasoning capability. In contrast, fully path-based reasoning better reflects the nature of graph inference but suffers from extremely low sample efficiency. We propose AdaG-Reason, a path-based LLM-KGC framework. We formulate reasoning data synthesis as a non-stationary multi-armed bandit problem and introduce Active Reasoning Distillation to select informative trajectories by jointly considering distribution alignment, relation difficulty, and uncertainty. We further extend this bandit-based sampling principle to the reinforcement learning phase, adaptively allocating rollout queries by per-relation difficulty. For training, we propose Implicit Reasoning Distillation optimized via Step-wise Group Relative Policy Optimization (GRPO), enabling direct and stable learning of pure graph navigation policies without value function estimation. Experiments on YAGO3-10 and NELL-995 show that AdaG-Reason achieves competitive MRR and state-of-the-art Hits@1 on YAGO3-10, demonstrating the effectiveness of active data selection and pure policy learning for path-based LLM-KGC.