The native Falklander's chances of long-term survival are not very good. The Falklands wouldn't support a very large population. Perhaps one island in the Falklands could be turned into a reservation, and the survivors of disease/massacre could be exiled there by the colonial government. It's still their homeland so they have a good chance of surviving in that environment, and being on an island makes controlling who lands there easy, so if inclined the government in the Falklands could prevent whalers, slavers, and other undesirables from attacking the native Falklanders.
The Falklands aren't really very good land, more useful for their strategic location near the Tierra del Fuego than anything else, so this does mean that there will not be too many land-hungry settlers seeking to break their home government's law to take Indian lands. A reservation system could preserve the Falklanders, but it's their only hope.