Advancements in large language models (LLMs) are revolutionizing interactive game design, enabling dynamic plotlines and interactions between players and non-player characters (NPCs). However, LLMs may exhibit flaws such as hallucinations, forgetfulness, or misinterpretations of prompts, causing logical inconsistencies and unexpected deviations from intended designs. Automated techniques for detecting such game bugs are still lacking. To address this, we propose a systematic LLM-based method for automatically identifying such bugs from player game logs, eliminating the need for collecting additional data such as post-play surveys. Applied to a text-based game DejaBoom!, our approach effectively identifies bugs inherent in LLM-powered interactive games, surpassing unstructured LLM-powered bug-catching methods and filling the gap in automated detection of logical and design flaws.