Right, but neither country would willingly give it up. Would they?
No, they wouldn't.
I see a possibility, but it's kind of out there: at the time neither the UK or the USA wanted either Argentina or Chile to fully own the Strait of Magellan or Cape Horn, and indeed, it's one of the reasons it is divided OTL. Maybe they could take over Tierra del Fuego, and some enterprising zionist could push for making it a Jewish homeland.
This would, of course, make
no one happy, probably not even the Jews moving there (Tierra del Fuego was quite unforgiving at the time, it was even a penal colony for Argentina). Most of Patagonia is rather unforgiving in any case. Little else to do at the time except sheepherding and fishing.
The closest equivalent would probably be the Welsh Colonies in Chubut,
https://en.wikipedia.org/wiki/Y_Wladfa or the various German/Czech/Italian/Russian/Hungarian and a long etc. settlements in Chaco, Misiones and elsewhere. They eventually all assimilated into wider Argentina, mostly by being overtaken by Italian and Spanish inmigrations (seriously, a LOT of Italians came to Argentina... my great-grandparents included).
The Welsh, like many other inmigrant collectivities, were actually invited by Argentina to colonize those harsh territories. Maybe an open-minded Argentine president (quite hard to get that in those times
) could also invite Jews to colonize Patagonia, or maybe the Chaco or Misiones.
But to be truly independent, a Jewish Argentine colony would need extraordinary inmigration (most Jews who inmigrated to Argentina went to the cities from what I understand) and foreign backing not to be absorbed by the growing Argentine state (which probably won't see with good eyes them becoming independent). So that means no being landlocked... but the rich coasts of Buenos Aires and Rio Negro are NOT going to be given up, and the rest of Patagonia is too harsh to attract further settlers, as the Welsh found out. It would take great efforts to make such a colony prosper, and I doubt it would be self-sufficient to survive until, say, the discovery of oil (which isn't that much oil anyways). It would be a land of shepherds and fishermen.
Still, could be a very interesting story.
A Jewish colony in the Chaco or Misiones would face even greater challenges. Misiones is covered by tropical rainforest, and the Greater Chaco is covered by hot, dry (or floodable!) forest, the Impenetrable (just the name tells you a lot). But still, they had a lot of inmigrant collectivities, just ask the Mennonites in Paraguay
https://en.wikipedia.org/wiki/Mennonites_in_Paraguay or my own Trentine ancestors. It would be also interesting to see how a potential Jewish colony interacts with the native peoples, whatever in Patagonia or the North.