Satisfiability Modulo Theory (SMT) solvers are foundational to modern systems and programming languages research, providing the foundation for tasks like symbolic execution and automated verification. Because these solvers sit on the critical path, their correctness is essential, and high-quality test formulas are key to uncovering bugs. However, while prior testing techniques performed well on earlier solver versions, they struggle to keep pace with rapidly evolving features. Recent approaches based on Large Language Models (LLMs) show promise in exploring advanced solver capabilities, but two obstacles remain: nearly half of the generated formulas are syntactically invalid, and iterative interactions with LLMs introduce substantial computational overhead. In this study, we present Once4All, a novel LLM-assisted fuzzing framework that addresses both issues by shifting from direct formula generation to the synthesis of generators for reusable terms (i.e., logical expressions). Specifically, Once4All uses LLMs to (1) automatically extract context-free grammars (CFGs) for SMT theories, including solver-specific extensions, from documentation, and (2) synthesize composable Boolean term generators that adhere to these grammars. During fuzzing, Once4All populates structural skeletons derived from existing formulas with the terms iteratively produced by the LLM-synthesized generators. This design ensures syntactic validity while promoting semantic diversity. Notably, Once4All requires only one-time LLM interaction investment, dramatically reducing runtime cost. We evaluated Once4All on two leading SMT solvers: Z3 and cvc5. Our experiments show that Once4All has identified 43 confirmed bugs, 40 of which have already been fixed by developers.