The exploration-exploitation trade-off is central to sequential decision-making and black-box optimization, yet how Large Language Models (LLMs) reason about and manage this trade-off remains poorly understood. Unlike Bayesian Optimization, where exploration and exploitation are explicitly encoded through acquisition functions, LLM-based optimization relies on implicit, prompt-based reasoning over historical evaluations, making search behavior difficult to analyze or control. In this work, we present a metric-level study of LLM-mediated search policy learning, studying how LLMs construct and adapt exploration-exploitation strategies under multiple operational definitions of exploration, including informativeness, diversity, and representativeness. We show that single-agent LLM approaches, which jointly perform strategy selection and candidate generation within a single prompt, suffer from cognitive overload, leading to unstable search dynamics and premature convergence. To address this limitation, we propose a multi-agent framework that decomposes exploration-exploitation control into strategic policy mediation and tactical candidate generation. A strategy agent assigns interpretable weights to multiple search criteria, while a generation agent produces candidates conditioned on the resulting search policy defined as weights. This decomposition renders exploration-exploitation decisions explicit, observable, and adjustable. Empirical results across various continuous optimization benchmarks indicate that separating strategic control from candidate generation substantially improves the effectiveness of LLM-mediated search.