Send a row boat with one soldier and a pop gun and land in the East Falklands?
But what happens when the British send a yacht with two Royal Marines and a harpoon? The back and forth struggle over the Falklands would be long and bitter, no doubt.
More seriously (or at least as seriously as the POD allows), Argentina in tacit alliance with Japan would throw US/UK policy in Latin America into paranoid chaos, and I'd think there would be at least a few cool heads in Buenos Aires well aware of such consequences. Of course given that Japanese landings on the Falklands would probably be quite small and unsupported, we might instead see Argentina act "In defense of our lands and seas, and all the Americas..." or some such by booting out the Japanese themselves after an abrupt DOW. With Argentina in "protective custody" of the islands, and offering at least symbolic support against Japan, the UK might accept negotiations that transfers them to Argentina de jure, especially if it came with entry against the other Axis powers or coaling/fuel/basing rights leased freely or cheaply.
A good deal for Argentina, if Axis sympathies don't scuttle it. Lots of lend-lease goodies, the Islas Malvinas, presteige, and a good chance of coming out of WWII stronger and more prosperous then when it began. Hindsight of course, given the uncertainty of Allied victory, but Argentina has no real reason to concede the Falklands or much else to anyone even with a conditional peace to WWII.
This could go the other way, but we'd need an earlier POD that has Japan actively fishing for allies, trade, or diplomatic support in Latin America well before 1941, and quite probably 1939 for that matter. I don't know how you'd get there without alarm bells ringing in Washington or London, and it'd be very odd and unlikely to have such a Latin American front... as interesting as it would be.