The accurate simulation of tsunamis is generally deemed a computationally expensive task, since the multitude of effects usually requires a three-dimensional model. On the other hand, two-dimensional, depth-averaged systems like the shallow water equations can be simulated fairly quickly, but they only capture the tsunami wave itself and fail to capture any dispersivity, since it assumes only hydrostatic pressure. For a higher accuracy, we consider the H-BMSS-$\gamma$ system [Escalante et.\ al, 2019]. It is hyperbolized by applying a Generalized Lagrange Multiplier procedure, and it contains a component for non-hydrostatic pressure. For the H-BMSS-$\gamma$ system, we develop three methods to model the effect of a moving ocean floor: the first method assume an instantaneous undersea earthquake. The second method adds a non-hydrostatic pressure correction as a response to the sudden earthquake. Finally, the third method couples the ocean floor movement to the non-hydrostatic pressure itself. We simulate the H-BMSS-$\gamma$ model with all three methods in Samoa. For the numerical discretization using the ADER-DG (Adaptive DERivative Discontinuous Galerkin) method, for which we also suggest an optimized implementation which utilizes the intrinsic tensor product structure to simplify the computational effort. Also, we avoid a nonlinear integration in the corrector step. Furthermore, we add support for adaptive mesh refinement. Finally, we show the high-order convergence of our model and the implementation, and we benchmark the three earthquake coupling methods which we derived. In particular, we see that all of them manage to capture dispersive effects.