Industrial process control demands policies that are interpretable and auditable, requirements that black-box neural policies struggle to meet. We study an LLM-driven heuristic synthesis framework for hot steel rolling, in which a language model iteratively proposes and refines human-readable Python controllers using rich behavioral feedback from a physics-based simulator. The framework combines structured strategic ideation, executable code generation, and per-component feedback across diverse operating conditions to search over control logic for height reduction, interpass time, and rolling velocity. Our first contribution is an auditable controller-synthesis pipeline for industrial process control. The generated controllers are explicit programs accessible to expert review, and we pair them with an automated audit pipeline that formally verifies key safety and monotonicity properties for the best synthesized heuristic. Our second contribution is a principled budget allocation strategy for LLM-driven heuristic search: we show that Luby-style universal restarts -- originally developed for randomized algorithms -- transfer directly to this setting, eliminating the need for problem-specific budget tuning. A single 160-iteration Luby campaign approaches the hindsight-optimal budget allocation derived from 52 ad-hoc runs totalling 730 iterations.