Have them go through customs every time they leave their property to get groceries. Get yourself a conscientious officer who makes sure he doesn’t have enough hours in a day to fetch something he forgot on his first run. Btw, tough luck with your utilities, I assume it’s quite a lot of paperwork for international piping and electrical connections, especially if your utilities company is motivated not losing 100$ a month.
https://en.m.wikipedia.org/wiki/Territorial_claims_in_Antarctica
Pretty sure the unclaimed chunk is big enough to hold most of them
That doesn’t solve the problem if they get in trouble somewhere else. Also they’d be bothering future Americans wherever we send them