This study proposes a surrogate homogenization model at a finite strain using a Radial Basis Function (RBF) interpolation. Recently, data-driven methods for computational homogenization have been studied to overcome the difficulty of conventional multiscale analysis methods, e.g., FE2. In particular, a variety of data-driven approaches have been developed for history-dependent materials such as elastoplastic composites, but to the best of our knowledge, most of the studies are based on neural networks (NNs). Although there is no doubt about the power of NNs, the input-output relationship is a black box, not allowing for our understanding of mechanical correspondence. To avoid such an invisible process, we focus on the RBF interpolation-based surrogate computational homogenization, which replaces the microscopic analysis for small-strain elastoplastic composites. The simple structure of RBF interpolation allows us to easily understand the input-output relationship. To improve the applicability to practical problems, we extend the model to the finite strain framework. Also, the macroscopic elastic degradation behavior caused by the microscopic damage is originally introduced into the surrogate model in this study. Specifically, to create a constitutive database, a series of numerical material tests are carried out on a representative volume element model composed of multiple elastoplastic materials involving damage. Then, RBF interpolation is performed using the combination of data points in the database to construct a surrogate homogenization model. Next, a macroscopic finite element analysis is conducted using the obtained model, followed by a localization analysis. Representative numerical examples are presented to discuss the validity of the proposed method.