Simulation-driven design of soft robots requires accurate and stable simulations. In this article, we present an inverse finite element approach enabling simulation-ready characterization of soft robotic materials from uniaxial test data. Representing test specimens with finite elements, and modeling the specimen-device coupling, we enable the simultaneous fitting of hyperelastic parameters to single or multiple tests performed on specimens of varying shape and size. To safeguard against simulation instabilities, or non-physical behavior, we reparameterize and bound parameters using the consistency with linear elasticity. We use our characterization on commonly used silicones, and discuss the resolution- and order-dependence of fitted parameters, and the Mullins effect in the context of simulation-based design.