Prediction and estimation of internal battery states are important tasks for safe operation of batteries. However, due to inherent uncertainties like parameter, model structural and measurement uncertainties, it is especially challenging to make accurate predictions. We present a novel method for handling the structural error of a thermo-electrochemical battery model. With structural error, we refer to the errors caused by simplifications taken during modeling. We extend the battery model of a LiFePO4-graphite lithium-ion cell with an internal, stochastic error model in a minimally-intrusive way. We find the optimal error model parameters with Approximate Bayesian Computation and compare two error models of different complexity: an auto-regressive and a white-noise multiplier for the heat source term. Both extended models are then used together with a particle filter for data assimilation to determine adequate uncertainty bounds for predictions of the surface temperature at a 1C discharge rate. We show that the auto-regressive extended model can use the assimilated data more effectively to increase the predictive performance of the model on average when compared to the white-noise extended model. Our main conclusion is that accounting for the time-correlated character of model errors helps improve data assimilation and predictive performance of physically-based thermo-electrochemical battery models.