Transformers usually expose one inference cost per trained model, while deployed systems often need multiple cost-quality operating points. We study Budgeted Attention Allocation, a monotone head-gating mechanism conditioned on a requested attention budget. Dense warm-starting is important for stability: on a robust synthetic sequence task, one budgeted model reaches 99.7% accuracy at 0.303 estimated attention cost and 100.0% accuracy at 0.504 cost. On held-out AG News with a custom word-level transformer, hard-gate adaptation turns soft cost control into measured single-thread CPU speed, reaching 82.1% accuracy with 1.28x speedup at budget 0.50. In pretrained BERT-Mini AG News, budgeted structural pruning reaches 87.6% accuracy with 1.20x speedup at budget 0.50; a validation-ranked zero-shot dense post-hoc structural baseline reaches 86.1%, and one recovery epoch raises that per-budget specialist to 87.9%. On DBpedia14, BERT-Mini budgeted gates reach 97.4% at exact budget 0.50 versus 96.6% for dense full attention. Static fixed-budget gates and recovered dense specialists remain strong. The contribution is therefore not universal dominance, but a reproducible feasibility study of one controllable checkpoint across budgets that can trade attention cost for accuracy and be converted into measured structural speedups on small CPU benchmarks.