Black-box optimization is a fundamental tool in science and engineering for optimizing objectives when gradient information is unavailable. It becomes especially difficult when the objective function is expensive to evaluate, limiting the evaluation budget to a few tens or hundreds of queries, and when good solutions occupy complex, low-measure regions of the search space. Generative models can supply useful structural priors in such settings, but existing generative BBO approaches bring significant evaluation cost. We identify three design principles for generative optimization under such low-budget conditions: avoid objective learning, optimize in candidate space, and make every evaluation count. Together, these principles motivate separating structural modeling from objective-driven search. We instantiate them in SPARROW, a simple sequential optimizer that maintains a persistent, ranked archive of evaluated candidates and uses a fixed, unconditional generative sampler solely as a corruption-refinement operator. SPARROW requires only access to the sampler's corruption and refinement processes, and never needs to evaluate the objective to train or guide it. Across three complementary settings, probing thin feasible geometry, disconnected high-performing regions, and failure-prone evaluations, SPARROW outperforms classical and generative baselines under strict evaluation budgets. These results demonstrate that separating structural priors from objective-driven search can be effective when evaluations are scarce and the search geometry is challenging.

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