Recent advancements in additive manufacturing and topology optimization enabled the design of complex lattice structures with applications across various industries, including aerospace, automotive, or chemical engineering. However, the computational cost of solving these structures using the Finite Element Method (FEM) can be prohibitive, particularly when dealing with large-scale systems arising from 3D geometries. The high memory and time requirements of direct solvers make them unsuitable for such problems, leaving efficient iterative solvers as the only viable alternative. To address these computational challenges, Reduced Order Models (ROMs) emerged as a promising solution, offering significantly faster simulations at the cost of some loss in accuracy. In this study, we propose a novel approach by integrating the multiscale ROM Empirical Interscale Finite Element Method (EIFEM) [1] as a preconditioner within the conjugate gradient iteration. The strategy can be viewed as an alternative to the two-grid V-cycle for cellular structures, where the coarse mesh is replaced by EIFEM [2]. In addition, we introduce a parametrized version of EIFEM that enables the efficient treatment of non-periodic structures and that can be incorporated into optimization algorithms for the design of lattice materials. We present a series of numerical experiments to validate the performance of this approach, demonstrating its potential to enhance the efficiency of numerical simulations in real-world applications.