Hybrid quantum systems are highly promising platforms for addressing important challenges of quantum information science and quantum sensing. Their implementation, however, is technologically non-trivial, since each component typically has unique experimental requirements. Here, we work towards a hybrid system consisting of a superconducting on-chip microwave circuit in a dilution refrigerator and optically trapped ultra-cold atoms. Specifically, we focus on the design optimization of a suitable superconducting chip and on the corresponding challenges and limitations. We unfold detailed microwave-cavity engineering strategies for maximized and tunable coupling rates to atomic Rydberg-Rydberg transitions in $\mathrm{^{87}Rb}$ atoms while respecting the boundary conditions due to the presence of a laser beam near the surface of the chip. Finally, we present an experimental implementation of the superconducting microwave chip and discuss the cavity characteristics as a function of temperature and applied dc voltage. Our results illuminate the required consideration aspects for a flexible, tunable superconductor-atom hybrid system, and lay the groundwork for realizing this exciting platform in a dilution refrigerator with vacuum Rabi frequencies approaching the strong-coupling regime.