Property-driven inverse design of hyperelastic microstructures is of great importance for engineering applications, as it enables the development of materials that achieve desired mechanical responses under diverse loading conditions. However, the high computational and experimental cost of forward evaluations significantly limits the feasibility of inverse design strategies, especially in the presence of non-linear material behavior. Data-driven surrogate models offer an efficient alternative, but typically require large amounts of labeled data, which are again time- and resource-intensive to obtain. In this context, Bayesian approaches provide a powerful framework by enabling uncertainty-aware surrogate modeling through active learning, thereby minimizing the required number of labeled samples. Moreover, inverse design problems often involve vast and diverse design spaces for which direct geometric parameterization is impractical. As a result, compact low-dimensional representations of microstructures become essential to efficiently explore the design space using a limited set of meaningful descriptors. In this work, we propose a unified inverse design pipeline consisting of three main steps: (1) low-dimensional representation of microstructures using two-point statistical correlations combined with principal component analysis (PCA); (2) construction of a multi-output Gaussian Process surrogate model to infer effective hyperelastic strain energy functions from standard mechanical experiments, enhanced by uncertainty-based active learning to reduce data requirements; and (3) an uncertainty-aware brute-force inverse design strategy to identify candidate microstructures whose responses are closest to a prescribed target behavior. The proposed framework is demonstrated through the inverse design of stochastic metamaterial films exhibiting orthotropic effective hyperelastic responses.