Mutual information (MI) is a commonly adopted utility function in Bayesian optimal experimental design (BOED). While theoretically appealing, MI evaluation poses a significant computational burden for most real world applications. As a result, many algorithms utilize MI bounds as proxies that lack regret-style guarantees. Here, we utilize two-sided bounds to provide such guarantees. Bounds are successively refined/tightened through additional computation until a desired guarantee is achieved. We consider the problem of adaptively allocating computational resources in BOED. Our approach achieves the same guarantee as existing methods, but with fewer evaluations of the costly MI reward. We adapt knapsack optimization of best arm identification problems, with important differences that impact overall algorithm design and performance. First, observations of MI rewards are biased. Second, evaluating experiments incurs shared costs amongst all experiments (posterior sampling) in addition to per experiment costs that may vary with increasing evaluation. We propose and demonstrate an algorithm that accounts for these variable costs in the refinement decision.