Artemis II did not contribute sufficient gravitational forces to be relevant.

The limit that the 3 body problem presents is that there is no exact solution, and inexact solutions will eventually desynchronize so much as to become worthless for predictions. But the key word is 'eventually' - if your initial measurements are good, there could be years (and in the case of the planets of the solar system, millions of years) worth of useful predictions.

That's not what the three body problem refers to.