Established finite strain plasticity models typically rely on specific free energy formulations that facilitate the time integration of the internal variables, often employing logarithmic strain measures. In contrast, many numerical frameworks describe the hyperelastic response using more general constitutive formulations based on the right and left Cauchy--Green tensors, e.g., the St.~Venant--Kirchhoff or the Neo--Hooke constitutive laws. Consequently, the consistent extension of these frameworks to incorporate inelastic deformation --- particularly rate-independent plasticity and viscoplasticity --- along with the associated numerical challenges remain insufficiently explored in the literature. To address this gap, we propose an elastic--viscoplastic constitutive model with a general free energy formulation expressed in terms of the right Cauchy--Green tensor. Depending on the chosen flow rule and hardening law, two main numerical challenges emerge during the return mapping procedure. First, evaluating large plastic strain increments poses difficulties for the exponential map integrator. Second, negative plastic strains may arise in individual iterations of the Newton--Raphson scheme, leading to non-physical and often non-evaluable states. We propose several strategies to address these issues and enhance numerical robustness. A logarithmic transformation of the evolution equations is employed to effectively manage large plastic strain increments. While the elastic predictor still verifies the occurrence of plastic flow, it no longer serves as the initial guess for the Newton--Raphson scheme when the corresponding plastic strain increment is numerically non-evaluable. Instead, an adapted predictor is constructed via interpolation between two mechanical states, namely the elastic predictor and the isochoric plastic predictor, using an in-house developed structure-preserving tensor interpolation scheme. Additionally, we augment the Newton--Raphson scheme with a backtracking line search, controlling the step size to mitigate the aforementioned numerical challenges. Due to their generality, some of these strategies may also improve the numerical robustness of established plasticity and viscoplasticity models.