The nonlinear shallow water equations are one of the most popular depth-averaged water wave systems for tsunami modeling. They can be derived from the incompressible Euler equations with free surface under the assumption that the characteristic length of the waves is much greater than the water depth, which is valid for a broad range of tsunamis, especially those generated by earthquakes. Moreover, the energy dissipated by shocks using this model has been observed to approximate well the breaking of water waves and, thus, their coastal run-up. For these reasons, great efforts have been made to develop large-scale, robust, and efficient numerical schemes for this system. GeoClaw is a collection of algorithms for the discretization of shallow flows over complex topographies with a wave-propagation formulation of Godunov-type methods and adaptive mesh refinement. It takes advantage of a well-balanced and positivity-preserving Riemann solver for the shallow water equations, capable of robustly handling wet-dry fronts. The shallow water equations neglect dispersive effects, which may be important in modeling tsunamis that involve shorter-wavelength perturbations. On the other hand, dispersive water wave models lack an inherent wave-breaking mechanism and are often formulated as systems of PDEs with high-order and mixed time-space derivatives, which requires the implicit inversion of differential operators. In recent years, hyperbolic approximations of such systems have been proposed as potentially advantageous, mainly due to the availability of explicit shock-capturing numerical schemes for such first-order systems. This work is concerned with the implementation, in GeoClaw, of a hybrid solver that transitions from a hyperbolic-dispersive system to the shallow water equations as required by the conditions of the problem. In particular, we consider the hyperbolic relaxation of a dispersive depth-averaged Euler system, presented by Escalante et al. We evaluate the performance of this solver with standard benchmarking tests and real tsunami data, and run simulations of hypothetical scenarios.