In this article we study a semi-discrete numerical scheme for the Maxwell-Klein-Gordon equation in two spatial dimensions, based on Finite Elements and Lattice Gauge Theory (LGT). The discretization procedure from LGT, together with a numerical quadrature, ensures gauge invariance of the scheme. The gauge invariance implies that the scheme is constraint preserving. We combine this with energy conservation to prove convergence of the scheme towards a weak solution, for initial data of finite energy. Finally, we propose and implement a numerical scheme to assess our results.