You would need the people who reached the island to actually bother sticking around/bringing relative in. But otherwise it wouldn't be hard to create a "Malvinian" culture. The Fuegians built good boats, and perhaps with a bit of prodding could end up building even better boats (to access better stocks of fish, better whaling, etc). Transfer from the Polynesians (via Rapanui who settled Juan Fernandez who then went to the mainland) is possible too, but likely to be a bit too late in history.
They'd likely transplant trees from the mainland and learn to shelter them with windbreaks to encourage growth. Otherwise they could build skin boats like the Aleuts and Inuit did--similar conditions and environment. The Falklands are rich in seals, seabirds, and of course fishing and whaling will harvest a lot of food. I believe it has similar plants to Tierra del Fuego.
I think it's reasonable the islands could support a population of a few thousand hunter gatherers, who will look similar to their mainland relatives. Some will likely go to South Georgia (which will be the limits of their ability to grow trees) and beyond to the South Sandwich Islands. There's a marginal chance they might go further south, seeking seals, whales, etc. with their skin boats. They could navigate along the ice sheet to the islands in the area (i.e. South Orkney) and eventually settle the Antarctic Peninsula. That could take a few thousand years and much cultural evolution to be able to settle that far.
Inevitably they'll be colonized but the islands are rather remote and have no resources besides sealing and whaling. So it could go to anyone who had interests there, be it France, Britain, Spain, etc. The locals will be settled in missions where their numbers will slowly decline to extinction along with their language, just like the OTL Fuegians.