The viscous froth model for two-dimensional (2D) dissipative foam rheology is combined with Marangoni-driven surfactant redistribution on a foam film. The model is used to study the flow of a 2D foam system consisting of one bubble partially filling a constricted channel and a single spanning film connecting it to the opposite channel wall. Gradients of surface tension arising from film deformation induce tangential flow that redistributes surfactant along the film. This redistribution, and the consequent changes in film tension, inhibit the structure from undergoing a foam-destroying topological change in which the spanning film leaves the bubble behind; foam stability is thereby increased. The system's behaviour is categorized by a Gibbs-Marangoni parameter, representing the ratio between the rate of motion in tangential and normal directions. Larger values of the Gibbs-Marangoni parameter induce greater variation in surface tension, increase the rate of surfactant redistribution and reduce the likelihood of topological changes. An intermediate regime is, however, identified in which the Gibbs-Marangoni parameter is large enough to create a significant gradient of surface tension but is not great enough to smooth out the flow-induced redistribution of surfactant entirely, resulting in non-monotonic variation in the bubble height, and hence in foam stability.