We report high-resolution single-crystal inelastic neutron scattering measurements on the spin-1/2 antiferromagnet Ba(TiO)Cu₄(PO₄)₄. This material is formed from layers of four-site "cupola" structures, oriented alternately upwards and downwards, which constitute a rather special realization of two-dimensional (2D) square-lattice magnetism. The strong Dzyaloshinskii-Moriya (DM) interaction within each cupola, or plaquette, unit has a geometry largely unexplored among the numerous studies of magnetic properties in 2D Heisenberg models with spin and spatial anisotropies. We have measured the magnetic excitations at zero field and in fields up to 5 T, finding a complex mode structure with multiple characteristic features that allow us to extract all the relevant magnetic interactions by modeling within the linear spin-wave approximation. We demonstrate that Ba(TiO)Cu₄(PO₄)₄ is a checkerboard system with almost equal intra- and interplaquette couplings, in which the intraplaquette DM interaction is instrumental both in enforcing robust magnetic order and in opening a large gap at the Brillouin-zone center. We place our observations in the perspective of generalized phase diagrams for spin-1/2 square-lattice models and materials, where exploring anisotropies and frustration as routes to quantum disorder remains a frontier research problem.