Hybrid-thinking language models expose both cheap non-thinking and expensive thinking modes, creating a deployment problem: deciding when additional reasoning is worth its token cost and override risk. We formulate this problem as budgeted opportunity allocation, where a controller first obtains a cheap seed answer, estimates the marginal value and risk of paid thinking, and preserves the seed unless an adoption gate accepts the thinking result. We propose Bora-Budget, a budget-aware inference framework for frozen LLMs, and instantiate it as Bora-Switch-Calibrated. The controller uses seed-state features to predict helpfulness, harmfulness, and thinking cost, then allocates thinking through either frontier operating points or an AutoThreshold policy calibrated to a total token budget. Across LLM scales on math dataset, the method substantially improves over random allocation and approaches uniform-thinking accuracy under lower token budgets. Transfer results on multiple-choice, code, and hard-QA tasks show promising but uneven generalization: trace length remains a strong baseline, harmful overrides are reduced but not eliminated, and learning how much to think remains open. These results position Bora-Budget as a seed-preserving control layer for budgeted inference rather than a replacement for stronger always-thinking baselines.