This paper addresses the overwhelming computational resources needed with standard numerical approaches to simulate architected materials. Those multiscale heterogeneous lattice structures gain intensive interest in conjunction with the improvement of additive manufacturing as they offer, among many others, excellent stiffness-to-weight ratios. We develop here a dedicated HPC solver that benefits from the specific nature of the underlying problem in order to drastically reduce the computational costs (memory and time) for the full fine-scale analysis of lattice structures. Our purpose is to take advantage of the natural domain decomposition into cells and, even more importantly, of the geometrical and mechanical similarities among cells. Our solver consists in a so-called inexact FETI-DP method where the local, cell-wise operators and solutions are approximated with reduced order modeling techniques. Instead of considering independently every cell, we end up with only few principal local problems to solve and make use of the corresponding principal cell-wise operators to approximate all the others. It results in a scalable algorithm that saves numerous local factorizations. Our solver is applied for the isogeometric analysis of lattices built by spline composition, which offers the opportunity to compute the reduced basis with macro-scale data, thereby making our method also multiscale and matrix-free. The solver is tested against various 2D and 3D analyses. It shows major gains compared to black-box solvers; in particular, problems of several millions of degrees of freedom can be solved with a simple computer within few minutes.