The sun’s pull on the ship at the outside limit at the moon, is obviously negligible compared to the pull of the Earth. This should be obvious, because otherwise the moon would have left long ago.

Why is this so difficult to understand? Honestly I think that misleading NASA graphic did a lot of damage.

You throw in acceleration, which I never mentioned and doesn’t matter. The Artemis II never left Earth’s gravitional pull, the original issue was effectively what if it missed - and the answer is no big deal.