Health monitoring of highway bridges during vehicle passage is essential for prolonging structural service life. Numerous vibration suppression strategies have been developed, including the installation of tuned mass dampers on bridges and the development of bridge-friendly vehicles through suspension tuning. The practical implementation of bridge-friendly vehicles may require transportation policies to ensure that the vehicles have optimal suspension parameters before crossing bridges. Previous studies have primarily focused on extending Den Hartog fixed-point theory using the time-freezing technique, by which tuned vehicle suspension parameters are determined from the time-varying transmissibility functions at the bridge midspan. In this approach, the optimal vehicle suspension stiffness is obtained by equalizing the transmissibility magnitudes at fixed points, while the optimal damping is determined by enforcing a zero gradient condition. This work employs a midspan tuning strategy to optimally tune the inerter-damper-assisted vehicle suspension for enhanced bridge vibration suppression. The inerter exhibits frequency-dependent behavior and an effective negative stiffness characteristic, producing forces that are out of phase by half a cycle with conventional springs under harmonic excitation. As a result, coupling the inerter with a viscous damper can significantly enhance energy dissipation. A demand-oriented design framework is proposed to determine the minimum inerter apparent mass by applying H_∞ norm optimization for bridge vibration suppression. The results indicate that superior bridge vibration suppression can be achieved when the inerter frequency exceeds the fundamental bridge frequency, since the inerter shifts the resonance frequencies to enable more effective bridge vibration suppression. The key physical parameters of the inerter-assisted vehicle suspension are determined, and both frequency- and time-domain responses are numerically simulated to validate the vibration suppression strategy. The proposed approach also provides insights for future studies on enhanced energy dissipation using electromechanical inerter-based systems.