The British didn't "discover" them - a number of countries (Spain, France, UK, USA etc) had people landed and living there but there was no claim nor any law . It got so bad the USN actually landed to knock heads together but refused to "administer" it. It fell into UK hands more or less by accident.
The Argentinean claim is just that - a baseless claim. Mexico has a better claim to Texas.
As for an Argentine invasion ... if you think the RN has declined since the 1980's the Argentinean Navy has gone off a cliff - no subs, no aircraft carrier, no modern aircraft, no money.
The only that would worry me about defending the FI is that you have one big base and one long runway - and no medium - long range anti-air missiles.