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This work presents a morpho-hydrodynamic model and a numerical approximation designed for the fast and accurate simulation of sediment movement associated with extreme events, such as tsunamis. The model integrates the well-established hydrostatic shallow-water equations with a transport equation for the moving bathymetry that relies on a bedload transport function. Subsequently, this model is discretized using the path-conservative finite volume framework to yield a numerical scheme that is not only fast but also second-order accurate and well-balanced for the lake-at-rest solution. The numerical discretization separates the hydrodynamic and morphodynamic components of the model but leverages the eigenstructure information to evolve the morphologic part in an upwind fashion, preventing spurious oscillations. The study includes various numerical experiments, incorporating comparisons with laboratory experimental data and field surveys.