Not sure if this helps, but if you form the Hispanic Alliance, you should be able to invite Cuba and the Dominican Republic into your faction. Even though those two countries aren't technically in South America, I was still able to get this achievement this way.
With Cuba and the Dominican in the fold, your 1940 Subs should have sufficient range (also hire the Navy advisor and Ship Designer that boosts range) to reach the capital of Iceland. Justify on Denmark, and declare war before Hitler invades Poland, so Denmark won't have joined the Allies yet. Once you occupy Reykjavik, the achievement should fire. It took me 25 1940 subs to get the necessary naval supremacy to activate the Naval Invasion.
Denmark was the only nation I attacked- I didn't have to fight the USA or anyone else. This was done on 1.7.1., so apologies in advance if this no longer works with the La Resistance DLC.