Studies of the Schwinger model in the Hamiltonian formulation have hitherto used the Kogut-Susskind staggered approach. However, Wilson fermions offer an alternative approach and are often used in Monte Carlo simulations. Tensor networks allow the exploration of the Schwinger model even with a topological θ-term, where Monte Carlo methods would suffer from the sign problem. Here, we study the one-flavour Schwinger model with Wilson fermions and a topological θ-term using Matrix Product States (MPS) methods in the Hamiltonian formulation. The mass parameter in this model receives an additive renormalization shift from the Wilson term. In order to perform a continuum extrapolation, the knowledge of this shift is important. We present a method suitable for tensor networks that determines the mass renormalization using observables such as the electric field density, which vanish when the renormalized mass is zero. Using this shift, the continuum extrapolation is performed for various observables.