We study, using micromagnetic simulations, the inelastic scattering of spin-wave beams on edge-localized spin-wave modes in a thin ferromagnetic film. In the splitting and confluence processes, the new spin-wave beams are generated with frequencies shifted by the edge-mode frequency. We report that inelastically scattered spin-wave beams in both processes not only change their direction of propagation but also undergo lateral shifts along the interface, analogous to the Goos-Hänchen effect known in optics. These shifts of inelastically scattered beams, for a few special cases described in the paper, can be in the range of several wavelengths, which is larger than the Goos-Hänchen shift of elastically reflected beam. Unexpectedly, at selected frequencies, we found a significant increase in the value of the lateral shifts of the scattered spin-wave beams formed in the confluence process. We show that this effect is associated with the cascading nonlinear processes taking place at the edge of the film and involving the primary edge spin wave. Our results make an important contribution to the understanding of the nonlinear nature of spin waves and provide a way to exploit it in signal processing with magnons.