Large language models (LLMs) can propose vaccine adjuvant designs with plausible mechanistic rationales, but such outputs often lack grounding in established immunological knowledge. We present SAVANT, a neuro-symbolic framework for LLM-based adjuvant design that combines two-stage constrained generation with literature-grounded verification. SAVANT first generates a natural-language design rationale, then renders it into an ontology-constrained domain-specific language (DSL) using logic masks and finite-state control. To reason under fragmented evidence and incomplete first-principles rules, SAVANT verifies generated designs against literature-derived symbolic evidence through minimal sub-theorem proof. In expert-aligned evaluation, SAVANT shows greater stability in stage-level rank alignment than the strongest LLM-judge baselines, while achieving the strongest alignment for immune-effect support and mechanistic reasoning. SAVANT provides graded, interpretable assessments that distinguish supported mechanisms, under-supported hypotheses, and knowledge gaps for hypothesis-driven experimentation. Data and code will be released at https://github.com/banjiuyufen/SAVANT.