The obvious solution is for the British to sell the island (for free while buying out the settlers) to the Tehuelche who then would prevent all Argentine access to the Islands and declare them an independent state. Obvious solution really.
I would be more in favor of the indigenous people of Argentina kicking the whites out of Argentina and relegating them to the Falklands.
Kicking indigenous people out of the country and relegating them to some island that has nothing to do with them, sounds pretty whack.
That, of course, would be the ideal solution. But do we really want Western Europe and North America to be full of Argentine refugees? Most Spaniards find them near insufferable as it is...