I am currently testing several VAR models to forecast inflation. All VAR models contain the same variables, the only thing that differs are the number of lags. Since I am forecasting, I reckon the pseudo out-sample-fit is most important (measured with MSE). Currently, VAR(4) and VAR(5) have the best pseudo out-of-sample-fit. But, the Granger-causality test show that the lags of VAR(4) do not all Granger-cause eachother, whilst the lags of VAR(5) do. Does this matter for the robustness of my VAR(4) model?
Since you are looking for a model to be used specifically for forecasting, focus on the properties that matter most for that specific purpose. Out-of-sample fit and AIC are the most popular direct measures of potential forecasting performance.
Meanwhile, Granger causality test tests whether a predictive relationship is absent in population. If we reject the null of absence of Granger causality, we tend to think it is present. However, we do not know how well we can estimate it in a given sample. If we cannot estimate it precisely enough, a poorly estimated relationship will hurt forecasting performance instead of boosting it. Here is where AIC and out-of-sample fit come in. If they show that a certain model performs better than another, you may expect that model to outperform also on the yet unseen data.
In conclusion, out-of-sample fit and AIC on the one hand and Granger causality test on the other hand serve different purposes. In your case, the former two are the relevant ones.