Taking theoretical considerations aside, Akaike Information Criterion is just likelihood penalized by the degrees of freedom. What follows, AIC accounts for uncertainty in the data (-2LL) and makes the assumption that more parameters leads to higher risk of overfitting (2k). Cross-validation just looks at the test set performance of the model, with no further assumptions.
If you care mostly about making the predictions and you can assume that the test set(s) would be reasonably similar to the real-world data, you should go for cross-validation. The possible problem is that when your data is small, then by splitting it, you end up with small training and test sets. Less data for training is bad, and less data for test set makes the cross-validation results more uncertain (see Varoquaux, 2018). If your test sample is insufficient, you may be forced to use AIC, but keepingkeep in mind what it measures, and what can assumptions it makescan make.
On another hand, as already mentioned in comments, AIC gives you asymptomaticasymptotic guarantees, and it's not the case with small samples. Small samples may be misleading about the uncertainty in the data as well.
