If you move your POD up a bit, if we start out with a slightly greater tendency for Inuit to hire on as locals and worker/seamen on whalers, I can see them getting to Antarctica in greater numbers during the heavy Antartctic whaling. Some may remain, or even bring their families.
Sorry, Antarctica is just too tough, even for the Inuit. During the little ice age, they basically had to abandon parts of the arctic as simply too barren.
You might have Finns or Lapps or Inuit or even Dene surviving on the near Antarctic Islands. But even there, it's not going to be easy. The Inuit were extremely good at surviving in a very specific kind of Arctic environment, even subtle changes in that environment meant life or death for whole populations.
So their very specific suite of cultural tools isn't a perfect or even a really good match. It would give them a chance, but mostly, we're talking dying time.