The MiniBooNE large axial mass anomaly has prompted a great deal of theoretical work on sophisticated Charged Current Quasi-Elastic (CCQE) neutrino interaction models in recent years. As the dominant interaction mode at T2K energies, and the signal process in oscillation analyses, it is important for the T2K experiment to include realistic CCQE cross section uncertainties in T2K analyses. To this end, T2K's Neutrino Interaction Working Group has implemented a number of recent models in NEUT, T2K's primary neutrino interaction event generator. In this paper, we give an overview of the models implemented, and present fits to published muon neutrino and muon antineutrino CCQE cross section measurements from the MiniBooNE and MINERvA experiments. The results of the fits are used to select a default cross section model for future T2K analyses, and to constrain the cross section uncertainties of the model. We find a model consisting of a modified relativistic Fermi gas model and multinucleon interactions most consistently describes the available data.