Machine Translation of Culture-Specific Items (CSIs) poses significant challenges. Recent work on CSI translation has shown some success using Large Language Models (LLMs) to adapt to different languages and cultures; however, a deeper analysis is needed to examine the benefits and pitfalls of each method. In this paper, we introduce the ChineseMenuCSI dataset, the largest for Chinese-English menu corpora, annotated with CSI vs Non-CSI labels and a fine-grained test set. We define three levels of CSI figurativeness for a more nuanced analysis and develop a novel methodology for automatic CSI identification, which outperforms GPT-based prompts in most categories. Importantly, we are the first to integrate human translation theories into LLM-driven translation processes, significantly improving translation accuracy, with COMET scores increasing by up to 7 points. The code and dataset are available at https://github.com/Henry8772/ChineseMenuCSI.