We present techniques for the multi-scale design of additively manufactured components made of sheet-network triply periodic minimal surface (TPMS) lattices with regards to stress and fatigue. The TPMS lattices considered include the Gyroid, Primitive, and IWP. The multi-scale design consists of determining the optimal topology of the macro component and the thickness field of the sheet-network TPMS within the component. The novelty of our method consists in the ability to incorporate stress and fatigue damage constraints in the optimization problem. The proposed methods rely on efficient surrogate models of the stress field in a unit cell of the TPMS \cite{nguyen2025}. To construct these surrogates, we demonstrate and exploit the symmetries of the stress field in the unit cell, whereby it is shown that it suffices to obtain the stress field in one of eight fundamental units of the unit cell. To perform the stress analysis of a macro component made of a TPMS, we employ a solid model in which the effective elasticity properties are a function of the local thickness. The components of the macro stress at a given point (e.g., an element centroid) are taken as the tractions acting on a unit cell and passed to the surrogate models, which return the stress field on the corresponding unit cell. Using the stress field, we can compute, for example, the largest von Mises stress or fatigue damage in the unit cell. Using established aggregation techniques in topology optimization, we can subsequently use these values to impose a constraint on the largest stress or damage in the component. The fatigue analysis considers non-proportional loading, and it uses critical plane methods to compute damage. We ascertain the validity of our methods via application to the L-bracket benchmark, and we demonstrate the effectiveness of the methods through multiple examples.