Two-dimensional shallow water models as implemented in TELEMAC-2D (www.opentelemac.org) are widely used for riverine and coastal applications. However, high-resolution finite element simulations remain computationally demanding in many-query settings, such as real-time forecasting, design optimization, and uncertainty quantification. This work proposes a hybrid projection-based model order reduction (MOR) framework for the shallow water equations, with emphasis on robustness and structure preservation in the presence of wetting and drying areas. Instead of embedding the reduced model into the legacy solver, a dedicated online finite-volume (FV) model is introduced. The FV model combines HLLC Riemann fluxes with the hydrostatic reconstruction, which ensures exact preservation of the lake at rest equilibrium and includes a positivity-preserving wet/dry treatment suitable for moving shorelines. During the offline stage, snapshots of the conservative variables (h, hu, hv) are collected from high-fidelity simulations and a mass-weighted POD basis is constructed using a diagonal mass matrix. A key difficulty is that typical time-marching POD-Galerkin models might lead to non-physical negative water depths, typically near wet/dry fronts and under basis truncation. To eliminate these violations without resorting to ad hoc clipping which fails to conserve mass, the correction is formulated as a constrained minimization problem in the reduced coordinate space. We seek the optimal reduced coefficients that minimize the distance to the predicted state while strictly satisfying cell-wise depth positivity and exact global mass conservation. The constrained problem is solved efficiently in the reduced space, enforcing positivity and exact mass conservation with minimal additional computational cost. The methodology is assessed on Thacker moving-shoreline benchmarks and additional tests involving dry regions. Results demonstrate stable long-time integration, accurate free-surface evolution, and systematic suppression of negative-depth events across reduced dimensions.