The rapid emergence of new entities -- driven by cultural shifts, evolving trends, and personalized user data -- poses a significant challenge for existing Speech Large Language Models (Speech LLMs). While these models excel at general conversational tasks, their static training knowledge limits their ability to recognize domain-specific terms such as contact names, playlists, or technical jargon. Existing solutions primarily rely on prompting, which suffers from poor scalability: as the entity list grows, prompting encounters context window limitations, increased inference latency, and the "lost-in-the-middle" phenomenon. An alternative approach, Generative Error Correction (GEC), attempts to rewrite transcripts via post-processing but frequently suffers from "over-correction", introducing hallucinations of entities that were never spoken. In this work, we introduce LOGIC (Logit-Space Integration for Contextual Biasing), an efficient and robust framework that operates directly in the decoding layer. Unlike prompting, LOGIC decouples context injection from input processing, ensuring constant-time complexity relative to prompt length. Extensive experiments using the Phi-4-MM model across 11 multilingual locales demonstrate that LOGIC achieves an average 9% relative reduction in Entity WER with a negligible 0.30% increase in False Alarm Rate.