We address the task of hierarchical multi-label classification (HMC) of scientific documents at an industrial scale, where hundreds of thousands of documents must be classified across thousands of dynamic labels. The rapid growth of scientific publications necessitates scalable and efficient methods for classification, further complicated by the evolving nature of taxonomies--where new categories are introduced, existing ones are merged, and outdated ones are deprecated. Traditional machine learning approaches, which require costly retraining with each taxonomy update, become impractical due to the high overhead of labelled data collection and model adaptation. Large Language Models (LLMs) have demonstrated great potential in complex tasks such as multi-label classification. However, applying them to large and dynamic taxonomies presents unique challenges as the vast number of labels can exceed LLMs' input limits. In this paper, we present novel methods that combine the strengths of LLMs with dense retrieval techniques to overcome these challenges. Our approach avoids retraining by leveraging zero-shot HMC for real-time label assignment. We evaluate the effectiveness of our methods on SSRN, a large repository of preprints spanning multiple disciplines, and demonstrate significant improvements in both classification accuracy and cost-efficiency. By developing a tailored evaluation framework for dynamic taxonomies and publicly releasing our code, this research provides critical insights into applying LLMs for document classification, where the number of classes corresponds to the number of nodes in a large taxonomy, at an industrial scale.