The shallow water equations (SWEs) are a first-order hyperbolic system used extensively to model free-surface geophysical flows, particularly in hazard assessment contexts. However, their inability to capture dispersive effects limits their accuracy in scenarios involving shorter-wavelength disturbances. Dispersive water wave models, while capable of representing such features, typically lack shock-forming mechanisms that are essential for approximating wave breaking. Additionally, these models often involve high-order and mixed space-time derivatives or differential constraints with no explicit time evolution, posing challenges for efficient large-scale computations. Recent works have proposed hyperbolic approximations to dispersive systems, which offer the potential for efficient, explicit shock-capturing numerical methods. In this work, we develop a hybrid solver that dynamically couples hyperbolic dispersive models with the SWEs across varying flow regimes. Our approach focuses on hyperbolic reformulations of a depth-averaged Euler system and the Serre–Green–Naghdi equations. We evaluate the solver's performance using both laboratory benchmarks and synthetic scenarios.