We introduce a new numerical strategy designed for the modelization and simulation of nonlinear interactions between surface waves in shallow water and a stationary partially immersed surface piercing object. At the continuous level, the flow domain is globally modeled with the nonlinear hyperbolic shallow-water equations, while the description of the flow beneath the object reduces to a nonlinear ordinary differential equation. The coupling between the flow and the object is formulated as a free-boundary problem, associated with the computation of the time evolution of the spatial locations of the air-water-object interface. At the discrete level, the proposed formulation relies on an arbitrary-order discontinuous Galerkin approximation, which is stabilized with a recent a posteriori Local Subcell Correction method. The time evolution of the air-water-object interface is computed from an Arbitrary-Lagrangian-Eulerian description and a suitable smooth mapping between the initial frame and the current configuration. For any order of polynomial approximation, the resulting global approximation algorithm is shown to: (i) preserve the Discrete Geometric Conservation Law, (ii) ensure the preservation of the water height positivity at the sub-cell level, (iii) preserve the class of motionless steady states (well-balancing), possibly with the occurrence of a partially immersed object. Several numerical computations, involving solitary waves and bores propagations and transformations over varying topographies and/or in the vicinity of the object, highlight that the proposed numerical model: (i) effectively allows to model the flow-object interactions and provides the time-evolution of the air-water-object contact points, (ii) is able to accurately handle strong flow singularities without any robustness issues, (iii) retains the highly accurate sub-cell resolution of discontinuous Galerkin schemes.