My top pick for said war that could have happened is the one over the Beagle Islands between Chile and Argentina. It was about to happen during Operation Soberania, but it was cancelled due to a combination of bad weather and the intervention of the pope. Had either of those things didn't happen (though I'm not sure how would the Argentine junta turn down the Papacy's offer), South America would have its share of a bloody conflict in the 20th century between two non-communist dictatorships that could have lasted at least several months, depending on how the US backs Santiago in spite of the ties between the countries plus that of Buenos Aires; not to mention possibly drawing in interest from the USSR and perhaps China given the Cold War setup.