To really convince a flat earther, the northern latitudes might require two trips, one during the winter and one during the summer. 24 hours of darkness and 24hrs of daylight at the same spot don't jive well with the model. 24 hours of darkness is hard to explain at the north as well with the flat earth model, but it's harder to convince yourself that something weird didn't happen because you can't tell that the sky reset right back to where it was 24 hours ago ('cause there's no frame of reference in the sky to follow).
The other thing about antarctica that's really difficult to explain is why features are so much closer together. If he does go there (I know he won't) he should look around at some of the bays and peninsulas that he passes while enjoying his steak and lobster at the cruise ship restaurant. The flat earth model requires them to be impossibly large compared to how they'll look.
I can't think of a more relaxing and enjoyable way to prove that one's entire worldview is nonsense.