The reconstruction of bathymetry from known free surface elevation via numerical solution of the shallow water equations (SWE) is an ill-conditioned and generally ill-posed inverse problem. Without proper regularization, a method may fail to converge, or steady-state topography may exhibit spurious oscillations even if a smooth exact solution is known to exist. The presence of noise in the data, or a poor choice of discretization techniques aggravates such issues. To filter out perturbations caused by ill-conditioning and/or presence of unresolvable fine-scale features, a numerical method for the inverse SWE problem must be equipped with carefully designed stabilization operators. In this work, we discretize the two-dimensional shallow water equations using continuous linear finite elements. Physical admissibility is enforced using a well-balanced and positivity-preserving algebraic flux correction (AFC) scheme. The underlying low-order discretization is a new extension of the algebraic Lax-Friedrichs method to the SWE system with non-flat topography. The high-order AFC version imposes inequality constraints on antidiffusive fluxes that recover the target scheme. A monolithic convex limiting strategy guarantees preservation of local bounds for the water height and velocity components. The steady-state bathymetry is calculated via time marching. Two approaches are used to cure the lack of well-posedness and avoid oscillations. The first one adds an artificial diffusion term to the conservation law for the water height. In the second approach, the regularization term consists of numerical fluxes that are constructed using a new optimal control method. An optimization problem is formulated for scalar flux potentials with the aim of minimizing the perturbation of the discretized shallow water equations and deviations from the measured free surface elevation. To suppress oscillations caused by non-smooth data, we use a total variation denoising approach. The first numerical results for one- and two-dimensional test problems are promising. These preliminary results include convergence studies for benchmarks with noise in the initial data and experiments with discontinuous bathymetry.