What if Argentina managed to enlist the military support of other South American countries in a bid to conquer the Falkland Islands? Could Britain resist such an Alliance, and how long for? What if for example Brazil threw in with Argentina?
The problem with this whole premise is you fail to understand how geopolitical or "state" like the various nations of South America act on daily basis. Although the news shows Ecuador and Peru becoming buddy or Chile and Argentina doing the same; at the same time those same nation would throw each other under the bus if they could get away with it. In essence there are still deep rooted and nationalistic elements to these nations, were ever nation has a issue with a least one of their neighbors. Venezuela for example has been accused of supporting the FARC in Colombia (I'm not saying they are, it just being reported). That is like the United States supporting the FLQ in Canada, or Canada supporting neo-nazis in the United States. Hell Venezuela and Ecuador were posturing war some few years ago against Colombia due to a airstrike against the FARC slightly within Ecuadorian territory by Colombia. Really friendly neighbors tend not to support or be willing to go into conflict with a allied/ friendly nation.
Bolivia and Peru both have serious bones to pick with Chile still. There is reason why the majority of Chilean military assets are up north in the arid regions, and its not because of the beautiful open spaces for training.... Peru and Ecuador both went to war only some twenty years ago over a border dispute, although it got resolved its debatable if both sides will ever be "friendly or courtesy" amongst each other. The list goes on and on, the short answer is the South America in the 21st century is general speaking extremely peaceful between states; numerous South America nations have have simmering issues that go back hundreds of years (i.e. the Argentina claim on the Falklands, Bolivia coast ext...). However while it is true that all nations of South America are becoming closer and closer together, (new S.America reserve bank, opening of borders, free-trade); this area due to demographic issues, corruption, a weak but promising democratic tradition, and disparity of wealth between the rich and power still have a long way to go.
In reference to your point about Brazil, you have to take into consideration what gain would a emerging power like Brazil get from supporting a war against a great power and permanent member of the UNSC; the short answer is nothing. Brazil is general speaking friendly amongst all nations, why would they ever "shoot themselves in the foot" by supporting a attack against the Falklands with material support. Any and all good will with Europe, Oceania (Australia and New Zealand), and likely North America (US and Canada a least) would totally evaporate over night. There military has no capability beyond South America and their littoral area, they're navy would be sunk as soon as they sortied by British Astute-class submarines. Which the Brazilian Navy currently has no similar ASW platform to engage such a threat (although they're developing assets). Furthermore how will all these South American nations insure the protection of critical overseas trade, and keep the sea lines of communication open against a nuclear armed power with serious overseas reach. There air force still flies upgraded but aging F-5s and their government is unwilling to invest in a ongoing program to get a proper 4.5 generation aircraft. There air to air refueling is based on 4 old Boeing C-137 Stratoliner, and two even older KC-130H Hercules. They're old slightly modern air-assets is twelve ex French Mirage 2000, that were meant only as a stop gap for a two years that have now served on for over eight years. The bulk of their ongoing operations are conduct by a very effective COIN aircraft the turboprop Super Tuncao, and forty three modernized sub-sonic AMX International AMX. While the AMX is capable of aerial refueling, it is very debatable if this aircraft have the necessary hardware/software/ pilot training to engage in over the horizon anti ship warfare, because bomb runs would be literally suicide against a Type 45 destroyer with its Sampson radar which can detect a golf ball five miles out. There most modern maritime patrol asset are five modernized P-3 Orion, which while a decent ASW platform would be hard pressed to find let alone destroy a Astute-class in ocean as large as the Atlantic. In conclusion Brazil Air Force and Navy is decent for their defence needs, but have virtual no blue-water or over the horizon engagement capabilities. And Brazil is likely socioeconomically and demographically the best able to funded a proper "blue-water" military within South America. Chile and Venezuela both have more highly developed militaries (modern F-16s, Leo 2, NASAMS for Chile, and Su-30, Hinds, T-80 for Venezuela) but are limited by their logistical assets, and economics to support larger militaries or decent operations on the other side of South America. In short the majority of militaries of South America are paper tigers none more so than Argentina. They (all of South America) lack effective C2 (command and control) ISTAR or data linked militaries, they don't have dedicated sat surveillance or communication, their fire-control systems on all their (land, maritime, air) systems lack the modern "smart" needs for guided needs all of which the UK has in mature platforms. Modern conflict are not about having a carrier (Brazil) or hundred of tanks (Chile and Peru) but how you can integrate your various assets into a "joint" hardened but flexible decentralized command scene. In many ways ISTAR capabilities in Afstan or Iraq are more important then when assets you can deploy against your enemy. Because if you know were your enemy is going, what his intentions are, and what he can bring into a engagement half the guessing of the enemy intent and capabilities for a in-field commander are gone, as a result the commander can deploy his force accordingly.
Final South America economy, military, and industrial sectors are not hardened against kinetic or non-kinetic strike, neither is the UK but their assets are a half a world away from the engagement area(s).
With all this said the UK is not doing the greatest either, demographic are taking their toll, they're economy is shrinking, and their military is the smallest since the Crimean War. Yet their nations producing trillions of dollars yearly, they're military has some of the most highly developed weapon systems in the world, and one of the only seven known nuclear powers. All in all you have to understand why South America nations while general supportive diplomatically to Argentina's claims to the Falklands, there are in no way or even close to supporting Argentina in a physical sense nor are their militaries.