The consistent coupled-mode theory (Athanassoulis & Belibassakis, J. Fluid Mech. vol. 389, 1999, p. 275) is extended and applied to the hydroelastic analysis of large floating bodies of shallow draught or ice sheets of small and uniform thickness, lying over variable bathymetry regions. A parallel-contour bathymetry is assumed, characterized by a continuous depth function of the form $h( {x,y}) {=} h( x )$, attaining constant, but possibly different, values in the semi-infinite regions $x {<} a$ and $x {>} b$. We consider the scattering problem of harmonic, obliquely incident, surface waves, under the combined effects of variable bathymetry and a floating elastic plate, extending from $ x {=} a$ to $x {=} b$ and $ {-} \infty {<} y{<}\infty $. Under the assumption of small-amplitude incident waves and small plate deflections, the hydroelastic problem is formulated within the context of linearized water-wave and thin-elastic-plate theory. The problem is reformulated as a transition problem in a bounded domain, for which an equivalent, Luke-type (unconstrained), variational principle is given. In order to consistently treat the wave field beneath the elastic floating plate, down to the sloping bottom boundary, a complete, local, hydroelastic-mode series expansion of the wave field is used, enhanced by an appropriate sloping-bottom mode. The latter enables the consistent satisfaction of the Neumann bottom-boundary condition on a general topography. By introducing this expansion into the variational principle, an equivalent coupled-mode system of horizontal equations in the plate region ($a {\leq} x {\leq} b)$ is derived. Boundary conditions are also provided by the variational principle, ensuring the complete matching of the wave field at the vertical interfaces ($x{=}a$ and $x{=}b)$, and the requirements that the edges of the plate are free of moment and shear force. Numerical results concerning floating structures lying over flat, shoaling and corrugated seabeds are presented and compared, and the effects of wave direction, bottom slope and bottom corrugations on the hydroelastic response are presented and discussed. The present method can be easily extended to the fully three-dimensional hydroelastic problem, including bodies or structures characterized by variable thickness (draught), flexural rigidity and mass distributions.