The fluid-mediated snap-through of a deformed elastic structure in a constrained channel involves a complex interplay of geometry, fluid inertial effects, amplification of the transient pressure, and transport phenomena in a strongly nonlinear fashion. In this paper, we derive a simple reduced model of an elastic sheet squeezed into a closed channel between upstream and downstream fluid compartments. The reduced theory involves an antisymmetric order parameter, a fourth-order potential with the curvature switching sign at the critical pressure, the effective inertia including both structural and hydrodynamic added-mass contributions, and a quadratic closure for transport based on reflection symmetry. The comparisons of the reduced theory with the exact solution of Oshri et al. reveal that the critical-pressure scaling, collapse of the near-threshold growth rate behavior under effective inertia, long-lasting presence at the asymmetric solution branch in the fluid-dominated limit, broadening of the transient pressure peak for small values of the sheet-to-fluid mass ratio, and crossing of the kinetic energy distribution are captured. This means that the closed-channel snap-through phenomenon can be physically interpreted within one reduced framework: the threshold selection, transient slowdown, delayed transport, and energy redistribution arise due to the same low-dimensional added-mass dynamics.