Recent advances in additive manufacturing push designers to investigate new and intriguing ideas. One of the most promising trends consists in mimicking nature where the distribution of materials and void is cleverly organized, leading to objects with optimal mechanical behaviors. However, this increasing interest in these so-called additively manufactured lattice structures (or architected structures) comes with new challenges, especially in the Design-to-Manufacturing loop. There is a known and crucial need for dedicated numerical methods for designing these structures with reasonable computational ressources and with convenient workflows. Here, the investigated workflow makes use of three key ingredients: IsoGeometric Analysis, Reduced Order Models, and Domain Decomposition methods. IGA offers CAD-based shape parameterization of the lattice structures, provides great analysis accuracy, and eases the construction of ROM. We take advantage of the natural decomposition of the computational domain in a set of similar cells by employing a DD solver which is further accelerated with pre-trained ROM. These ROM substitute the load/displacement relationships at cell-to-cell interfaces, leading to fast and memory efficient solver. Furthermore, we show how such a framework unlocks the possibility of performing the shape optimization of lattice structures at the full fine-scale of the structure (unlike homogenization-based approaches).