The Moon will not be visible in Union Glacier at all during the trip. It will have a Celestial Declination ranging between 14° to 29° meaning it is above the Tropic of Cancer whereas the Sun will have a Celestial Declination of -23° meaning it will be in your perspective since it’s above the Tropic of Capricorn at -23° latitude.
However at -75°S latitude on December 20th the Moon will appear in perspective at 1° above the horizon north. That’s the moment it begins to enter your perspective on Flat Earth and it will only be for a minute. You would need to keep going North to bring the Moon into your perspective. Greater arc lengths (longer period of Moon light) means you are bringing it closer and closer into full perspective where eventually the Lunar Zenith (when the Moon is directly above) would happen in this at latitude 14° since the Celestial Declination is also 14° on Dec 20th. Flat Earth math. 1:1 relationships. Very simple.
Therefore we know on Flat Earth if you move 5°S towards Union Glacier at 80°S the Moon disappears out of perspective.