The advent of Additive Manufacturing (AM), also known as 3D printing, is pushing the development of architected materials to a new stage. These multiscale heterogeneous lattice structures can be tailored at the level of the unit-cell to achieve specific performance requirements. As the computational cost and memory requirements related with their numerical simulation become quickly tremendous, the efficient prediction of their structural response is gaining considerable interest. In this work, we develop a fast solver for the full fine-scale analysis of gyroid-like metamaterials. We define the overall macro-structure implicitly by prescribing the coefficients of a parametrized porosity function within each cell. The geometry representation is then fully decoupled from the underlying discretization by the use of unfitted domain methods. Our main purpose is to exploit the natural domain decomposition into unit cells by designing an efficient two-level FETI-DP solver. Importantly, the local sub-structuring operators are approximated with projection-based reduced order models, which are built during the offline phase. In these regards, we rely on localized approaches, what allows to retain good accuracy despite the inherent discontinuities introduced by the unfitted discretization. This results in an accurate and scalable algorithm with significant reductions in the online computational cost. The solver is tested against various 2D and 3D examples; it shows major gains with respect to black-box solvers.