Recently, a novel, nonparametric, probabilistic method for modeling and quantifying model-form uncertainties in nonlinear computational mechanics was proposed. Its potential was demonstrated through several uncertainty quantification (UQ) applications in vibration analysis and nonlinear computational structural dynamics. This method, which relies on projection-based model order reduction in order to achieve computational feasibility, exhibits a vector-valued hyperparameter in the probability model of the random reduced-order basis and associated stochastic, projection-based reduced-order model. It identifies this hyperparameter by formulating a statistical inverse problem grounded in target quantities of interest and solving the corresponding nonconvex optimization problem. For many practical applications however, this identification approach is computationally intensive. For this reason, this paper presents a faster, predictor-corrector approach for determining the appropriate value of the vector-valued hyperparameter that is based on a probabilistic learning on manifolds. It also demonstrates the computational advantages of this alternative identification approach through the UQ of two three-dimensional, nonlinear, structural dynamics problems associated with two different configurations of a MEMS device.