Approximate nearest neighbor search (ANNS) is a critical component in modern data-intensive applications, but its performance is often hindered by the use of a static query budget parameter. This one-size-fits-all approach, even if well-tuned, fails to account for the varying difficulty of individual queries, inevitably leading to suboptimal latency on easy queries and poor accuracy on hard ones. This paper introduces QBAT, a query-aware budget autotuner designed to resolve this dilemma. By analyzing query-specific features offline, QBAT dynamically allocates an appropriate budget for each query. We explore two predictive models: a highly accurate gradient-boosted decision tree and a simple, interpretable heuristic formula derived using the AlphaEvolve framework. These models can optimize budget allocation for both system performance or recall consistency priorities. Evaluations on large-scale datasets demonstrate that QBAT reduces total searched budget by up to 68.8% in the consistency mode on ScaNN, the state-of-the-art clustering-based ANNS method, while simultaneously enforcing a strict per-query recall target, a scenario where static budgets are notoriously inefficient and wasteful.