In many scientific and engineering domains, physical experiments are often costly, non-replicable, or time-consuming. The Kennedy and O'Hagan (KOH) model framework has become a widely used approach for combining simulator runs with limited experimental observations. Under a Bayesian implementation, the simulator output, model discrepancy, and observation noise are jointly modeled by coupled Gaussian processes, followed by coherent posterior inference and uncertainty quantification. This work presents a genuinely sequential Bayesian experimental design (BED) framework explicitly aimed at improving the predictive performance of the KOH model. We employ a mutual information (MI)-based criterion and develop a hybrid variant that integrates it with measures of local model complexity, leading to significantly more efficient design decisions. We further show theoretically that the MI-based criterion is more comprehensive and robust than the classical integrated mean squared prediction error (IMSPE) minimization criterion, especially when the model is highly uncertain in the early stages of the experiment. To mitigate the computational burden of fully Bayesian inference and the ensuing BED process, we propose two acceleration strategies - Gaussian Mixture Compression and Schur complement and rank-one update - which together substantially reduce runtime. Finally, we demonstrate the effectiveness of the proposed methods through both a synthetic example and a real biochemical case study, and compare them against several classical design criteria under sequential (offline) and adaptive (online) BED settings.

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