Diffusion caching is a lightweight strategy that accelerates Diffusion Transformers (DiTs) by reusing intermediate features across denoising steps, but existing paradigms face a fundamental trade-off: online heuristics lack global budget awareness, whereas static schedules lack instance adaptivity and fail to flexibly adapt to varying runtime budget constraints. To bridge this gap, we present BAG (Budget-Aware Gating), a novel caching policy that unifies global budget pacing with dynamic, instance-adaptive feature reuse. Rather than relying on hand-crafted rules, BAG employs a lightweight gating network that dynamically decides whether to execute a full computation or reuse cached features at each step by jointly conditioning on the budget state and local trajectory feedback. We train this policy via offline-to-online schedule distillation, transferring the decision-making of offline-searched schedules into a compact online gate. Extensive experiments on FLUX.1-dev and Wan2.1 demonstrate that BAG consistently outperforms state-of-the-art caching methods across various speedup tiers while remaining robust across different resolutions, seeds, and guidance scales. Code will be released.