Ayurveda is one of the world's oldest systematized medical traditions, yet standard biomedical ontologies such as SNOMED CT and ICD lack Ayurvedic diagnostic constructs, creating barriers to interoperability and clinical decision support. We present AyurAssist, a clinical decision support system that bridges this terminological gap through a vocabulary-grounded pipeline: biomedical named entity recognition (scispaCy) extracts clinical entities from free-text patient narratives, UMLS normalizes them to SNOMED CT and ICD codes, and fuzzy matching against the full 3,550-term WHO International Terminology for Ayurveda (ITA) constructs a structured Ayurvedic context for a large language model (Qwen3-32B), which performs three-pass clinical reasoning. We validate the system through three complementary experiments. First, benchmarking on BhashaBench-Ayur (14,963 questions) establishes Qwen3-32B (54.2%) as the top-performing model, outperforming the domain-specific AyurParam-2.9B (40.0%). Second, an ablation study over 80 clinician-annotated vignettes demonstrates that the terminology bridge yields a statistically reliable improvement in treatment quality (term-level F1: 0.219 vs. 0.156; near-disjoint 95% bootstrap CIs) and a directionally consistent gain in diagnostic accuracy (80.0% vs. 75.0%), while the bridge alone achieves only 5.0%, confirming that the LLM performs clinical reasoning, while the bridge provides essential vocabulary grounding. Third, inter-rater reliability across four clinicians (two Ayurvedic, two allopathic) establishes ground-truth validity with substantial agreement for modern diagnosis (PABAK= 0.66) and moderate agreement for Ayurvedic diagnosis (PABAK= 0.56). The key insight is that domain-specific vocabulary grounding of a capable general-purpose LLM, rather than domain-specific model training, offers a practical and scalable path toward interoperable integrative medicine informatics.