Wait a year or so. Royal Navy loses the carriers (Invincible to Australia and Hermes to India) and the Vulcans leave service (possibly even sold to Argentina
https://www.express.co.uk/news/uk/603285/Plane-crazy-UK-plan-sell-Argentina-bombers) - no air support for the task force. War is either a damp squib as the British don’t go, or a slaughter if they do.
Great Britain agrees to give up all claims to the Falkland Islands peacefully. Otherwise without some major luck on the part of Argentina, Great Britain is going to win.
The problem with that is that it is very unlikely that the Argentine Junta would have lasted long enough to carry it out. The invasion was carried out to distract the public from troubles at home.
Of course I have an even longer term plan which is to continue as they were and eventually the Falklands will be handed over as a cost cutting measure by the British Government.
We sell Argentina a few dozen more Exocet so they can spam missiles at stand-off ranges against a pretty shitty anti-missile defence capability.
Someone less decisive than Thatcher at the helm might delay the response, perhaps, allowing the Argentinians to place more missiles on the islands themselves and in general put themselves into a better position?
The central issue with making Argentina win is that making the British task force fail is different than winning the war. Doing the former might be possible. Argentina had exactly five Exocets (a weapon system first introduced in 1973) during the entire conflict, so that's an obvious POD. They could have simply ordered 200 earlier than IOTL, and seeing as how of the three times they launched them they scored two ships sunk and one badly damaged, the results can be extrapolated. That could have forced the task force back, but what happens then? Britain has other options. Nuclear is probably off the table (though it would seem to be the ultimate trump card), but Britain still has a potent submarine force. They could, say, blockade the Argentinian coast around a declared exclusion area (and the instant they announced they were doing that, every ship insurer would cancel policies for anything in that area) and focus on using their subs to sink Argentina's warships. Given the UK's nuclear submarine force and the ability to stage out of St. Helena, they could logistically sustain it, and they would wipe the floor with the Argentinians that way. In the end, that would probably force Argentina into submission (remember, the whole invasion was launched to distract from their domestic economic troubles, and with sanctions and the blockade those would have come back with a vengeance once the euphoria wore off.
The really big thing, though, is U.S. intervention. Now, it was important for the UK to take the lead on defending its sovereign territory from a third world country for the sake of its international image and reputation and to prevent anyone else from getting similar ideas. Guatemala had been making noises about invading Belize pretty hard in late 1970s, and Iraq threatened to invade Kuwait (when it was really a British protectorate) in 1961, with further border skirmishes in the 1970s. If the world saw Argentina successfully carve a piece off of Britain's own sovereign territory, it might have emboldened some bad actors. But the bottom line is that if they proved unable to do the job, the U.S., especially under Reagan, was not going to allow a third rate power to carve off territory from a nuclear NATO superpower and hand their forces consistent battlefield defeats. That would have been TREMENDOUSLY destabilizing at a very dangerous time in human history. Reagan made it clear what the U.S. was going to do IOTL with his offer to hand over the
USS Iwo Jima if necessary.
If push really, really comes to shove, the U.S. basically launches Operation Urgent Fury in the Falklands instead of Grenada.
