No Latin American country, not even Brazil, has the power projection necessary to pull this off.
But do they have to? There is nothing these saying it cannot be supported by another power. Cuba did just this fighting the South Africans in Angola, using Soviet logistical support to get the job done. That might count against the "first world nation involved" part, but maybe not, if none of their guys are doing the fighting.
Juanml82's scenario is interesting. It would, however, require Argentina's carrier to have better airplanes, a toughie since Super Etendards and Skyhawks aren't gonna do much to stop a land-based air force. If we go far back, to WWII's aftermath, we can change that, I think. I was thinking that Brazil and Argentina, having been major supporters of the war effort (and in Brazil's case a belligerent), the United States decides to include them in the Marshall Plan, along with Mexico, which also contributed. Peron never comes to power or his influence is reduced to the point that Argentina and Brazil have stable governments and major economic growth in the 1950s and 1960s, leaving Argentina firmly in the first world camp by the oil crisis and Brazil most of the way there. Having been helped by the United States to such a level, they are both pro-West (maybe even NATO members) and maintain powerful armed forces. The Falklands never comes to be much of an issue, as Argentina would rather be friends with the United Kingdom than have the Islas Malvinas for themselves. Hence, both nations have much bigger economies and much stronger militaries by the 1990s.
As with the West, they are targeted by Islamic Terrorism. Hezbollah does the attack in Buenos Aires in 1992, and Argentina angrily demands Syria and Lebanon arrest those responsible. They refuse. Argentina sends an aircraft carrier and naval forces, including Marines, to Lebanon to make a statement. Syrian AF fighters attack this force, and the Argentines fight back. Angered, the Argentinian Marines land in Lebanon briefly, and Argentinian carrier aircraft attack Hezbollah positions in Lebanon.