As you mentioned, timing is a problem, and even positive trade relations wouldn't have wedged Argentina out of Britain's sphere fast enough to prevent that. What I'm looking towards instead now is the possibility of Argentina peeling themselves off of Britain, either with active German aid or not, by taking advantage of certain geopolitical shifts in the late 30's and early 40's which will keep Britain distracted and Germany more at the forefront.
The issue is that Brazil is a Latin-speaking, Catholic, modern Authoritarian nation, while Germany is pushing towards something which, on paper, is old-fashioned, at least in governance. I feel that Italy would be a more natural friend, especially since many of Italy's friends are also not conventionally fascist.
Another hot-button region I see is Venezuela, which is run by a military dictatorship, has enormous oil reserves, and which claims territory from its neighbours, notably the Dutch in the ABC Islands and British Guyana.
Aside from those, my knowledge of critical junctures and the influence of WW2 and its lead-ups IRL in the region is sadly lacking. The only other major factor I know in the region is the United Fruit Company, which, while undoubtedly affected by changes in the global market brought about by war, also wouldn't be massively different without some major divergences.