We use inverse methods of statistical mechanics to explore trade-offs associated with designing interactions to stabilize self-assembled structures against changes in density or temperature. Specifically, we find isotropic, convex-repulsive pair potentials that maximize the density range for which a two-dimensional square lattice is the stable ground state subject to a constraint on the chemical potential advantage it exhibits over competing structures (i.e., "depth" of the associated minimum on the chemical potential hypersurface). We formulate the design problem as a nonlinear program, which we solve numerically. This allows us to efficiently find optimized interactions for a wide range of possible chemical potential constraints. We find that assemblies designed to exhibit a large chemical potential advantage at a specified density have a smaller overall range of densities for which they are stable. This trend can be understood by considering the separation-dependent features of the pair potential and its gradient required to enhance the stability of the target structure relative to competitors. Using molecular dynamics simulations, we further show that potentials designed with larger chemical potential advantages exhibit higher melting temperatures.