In this paper, a geometrically nonlinear inerter (NI) device consisting of two inerters with a designed configuration is presented. Such a device is integrated in an inertial nonlinear energy sink (INES), and we examine its vibration suppression performance using the Runge-Kutta (RK) method, which is verified by the harmonic balance method (HBM). The new isolator is compared to a traditional vibration isolator, and the results show a significant improvement in vibration suppression performance. To investigate the effects of different excitation amplitudes and initial conditions on the dynamics of the system, a series of transmissibility-frequency response analyses are performed by defining the displacement transmissibility. These analyses are carried out at frequencies near the natural frequency. The energy flow of the system is analyzed, and numerous calculations reveal a series of ideal values for the energy sink in the NI-INES system. This study provides new insights for the design of vibration isolators.