Deformable mirrors (DMs) are increasingly becoming part of nominal coronagraph designs, such as the hybrid Lyot coronagraph, in addition to their role counteracting optical aberrations. Previous studies have investigated the effects of the inter-DM Fresnel number on achievable contrast, throughput, and tip/tilt sensitivity for apodized coronagraphs augmented with DMs to suppress diffraction from struts and segment gaps. In this paper, we build upon that previous work by directly suppressing tip/tilt sensitivity with the controller, both for coronagraphs with and without apodizers. We also explore the effects of other important design parameters such as actuator density and tip/tilt controller weighting on performance. These comprehensive coronagraph design studies are enabled by the Fast Linearized Coronagraph Optimizer (FALCO) software toolbox, which provides rapid re-calculation of the DM response matrix for a variety of coronagraphs.