The Poisson-Nernst-Planck-Navier-Stokes (PNP-NS) system models the transport of charged species in a fluid under electric fields, coupled with fluid flow dynamics. While the classical PNP-NS framework provides a foundational model, it has limitations in capturing complex electrokinetic phenomena and remains a topic of active research. To address these challenges, several modified formulations have been proposed in recent years. We briefly review some of these approaches and their limitations, then focus on a specific model based on the Landau-Ginzburg-type continuum theory for room-temperature ionic liquids (RTILs), which leads to a modified fourth-order PNP-NS system. The well-posedness of the model is first established at the continuous level. A semi-discrete numerical scheme is then introduced based on a conforming Virtual Element Method (VEM) discretization. The proposed scheme employs an $H^2$-conforming virtual element space for the Poisson equation, an $H^1$-conforming space for the Nernst-Planck (NP) system, and divergence-free conforming spaces for the Navier-Stokes (NS) equations. For time discretization, a fully implicit backward Euler scheme is adopted to ensure robustness and stability. Well-posedness of the resulting numerical scheme is proven, and error estimates are provided for both the semi-discrete and fully discrete formulations in different Bochner norms. Finally, numerical experiments are presented to validate the theoretical results and demonstrate the effectiveness of the proposed approach.