This paper is dedicated to the full discretization of linear Maxwell's equations, where the space discretization is carried out with a discontinuous Galerkin (dG) method on a locally refined spatial grid. For such problems explicit time integrators are inffcient due to their strict CFL condition stemming from the fine mesh elements in the spatial grid. In the last years this issue of so-called grid-induced stiffness was successfully tackled with locally implicit time integrators. So far, these methods were limited to unstabilized (central fluxes) dG methods. However, stabilized (upwind fluxes) dG schemes provide many benefits and thus are a popular choice in applications. In this paper we construct a new variant of a locally implicit time integrator based on an upwind fluxes dG discretization on the coarse part of the grid. In contrast to our earlier analysis of a central fluxes locally implicit method, we now use an energy technique to rigorously prove its stability and provide error bounds with optimal rates in space and time.