With the increasing maturity of additive manufacturing technologies, the fabrication of architected and lattice-based metamaterials has become feasible for industrial applications. These materials offer an attractive combination of lightweight design and highly tunable mechanical properties driven by their underlying architecture. Accurately predicting their mechanical response, however, remains a major numerical challenge. Fine-scale simulations require resolving thousands of geometrically complex unit cells, often without sufficient scale separation to enable classical multiscale homogenization techniques. In this work, we propose a dedicated solver for full isogeometric volumetric fine-scale simulation of linear and nonlinear lattice structures that significantly reduces both computational cost and memory requirements. The proposed approach exploits the intrinsic self-similarity of the unit cells through a reduced-order modeling (ROM) strategy embedded within a domain decomposition framework. By constructing reduced bases from a limited number of ”principal” cells, local stiffness operators can be accurately approximated in a low-dimensional space. This drastically decreases the number of local computations required during stiffness assembly and linear system solution, while enabling efficient formation and storage of the global stiffness matrix. An inexact ROM-based FETI-DP-based solver is first introduced. This method allows the solution of problems involving several thousand cells (corresponding to several million degrees of freedom) in only a few minutes on a standard laptop. Both small- and large-deformation regimes are addressed, for linear elasticity and nonlinear hyperelastic models. Finally, a parallel multilevel preconditioner extending this strategy is presented, enabling large-scale simulations on massively parallel architectures with up to several thousand processing cores.