Complex structural systems often entail computationally intensive models that require efficient methods for statistical model calibration due to the high number of required model evaluations. In this paper, we present a Bayesian inference-based methodology for efficient statistical model calibration that builds upon the combination of the speed in the computation of a low-fidelity model with the accuracy of the computationally intensive high-fidelity model. The proposed two-stage method incorporates the adaptive Metropolis algorithm and a Gaussian process (GP)-based adaptive surrogate model as a low-fidelity model. In order to account for model uncertainty, we incorporate a GP-based discrepancy function into the model calibration. By calibrating the hyperparameters of the discrepancy function alongside the model parameters, we prevent the results of the model calibration to be biased. The methodology is illustrated by the statistical model calibration of a damping parameter in the modular active spring-damper system, a structural system developed within the collaborative research center SFB 805 at the Technical University of Darmstadt. The reduction of parameter and model uncertainty achieved by the application of our methodology is quantified and illustrated by assessing the predictive capability of the mathematical model of the modular active spring-damper system.