Bayesian calibration has become a powerful tool for estimating uncertain parameters in complex engineering models. Yet in many practical situations, the data available for calibration is limited or not particularly informative, leaving substantial uncertainty in the inferred parameters and restricting the insights needed for reliable decision‑making. In many experimental settings, however, the data generation can be influenced through controllable design choices, for example, by selecting sensor locations, loading scenarios, or testing configurations. Importantly, these choices are often restricted to a discrete set of feasible designs rather than a continuous design space. In this presentation, we introduce an adaptive strategy for Bayesian optimal experimental design when only a finite set of candidate designs is available, and the underlying computational model is expensive to evaluate. The approach relies on an accelerated nested Monte Carlo scheme that reuses parameter samples to limit the number of required model evaluations. To further improve comparisons of expected information gain, the method incorporates common random numbers together with multiple noise realizations per parameter sample. A bootstrap-based procedure then quantifies the probability that one design provides greater information gain than another, without making assumptions about the underlying distribution. The algorithm begins with modest sample sizes and progressively eliminates inferior designs, increasing computational effort only for the most competitive candidates until a single design is identified. Our elimination framework enables highly efficient and reliable experiment selection, making it suitable for real-world engineering applications involving expensive simulations. We demonstrate this through various challenging numerical examples, such as the calibration of material parameters of a human lung model.