Elastomeric materials are widely used in the automotive industry, essential for noise, vibration and harshness control. Components such as bushings and mounts rely on elastomers due to their inherent visco-hyperelastic behavior, combining nonlinear elasticity with time- and frequency-dependent effects. While beneficial for vibration isolation, they complicate accurate modelling for system design and analysis, particularly when combined with complex component geometries, often requiring large-scale nonlinear finite element models (FEMs) with high computational cost. To accelerate FEM simulations with state-dependent nonlinearities, the commonly used model order reduction technique combines the reduced-basis method with hyper-reduction and have been applied to visco-hyperelastic structures [1]. However, this method typically requires costly quasi-static and transient full order model simulations and challenging snapshot selection for time dependent problems, and the resulting reduced order models still rely on the same iterative solvers, limiting their overall efficiency. Therefore, in this work, a model order reduction framework is proposed for efficient modelling and simulation of visco-hyperelastic systems, reducing model size, FE update and iterative computation. First, the model size is reduced by constructing projection bases via rational interpolation from viscoelastic model instances with fixed hyperelastic responses. This accelerates the reduced-order model construction compared to reduced-basis methods. Then, the asymptotic numerical method [2] is extended to visco-hyperelastic models, which avoids frequent FE updates. Consequently, the hyper-reduction is circumvented and the iterative computation is largely replaced with constant FE matrices, further saving the computation time. This methodology is demonstrated using an electric vehicle bushing component. [1] F Trainotti, J Marinko, J Maierhofer, DJ Rixen. ECSW hyperreduction of hyper-viscoelastic components via co-simulation with Abaqus. Finite Elements in Analysis and Design, 2024, 241, 104222. [2] M Potier-Ferry. Asymptotic numerical method for hyperelasticity and elastoplasticity: a review. Proceedings of the Royal Society A, 2024, 480(2285), 20230714.