I doubt that the US would have readily allowed any state of South America to unilaterally enter WW 2 on the side of Germany without severe reprecussions, and even possibly invasion. If you look at the political patterns early in the war this is clear.
Starting with the Declaration of Panama on Oct 3, 1939, about a month into the war at the first meeting of the Foreign Ministers of the American Republics where a "Hemispheric Safety Belt" of 300 to 1000 miles wide was declared from Canada to the southern tip of South America as an exclusion area to all belegerents. This excluded all combatants from this area for war purposes. Of course, all of the combatants also ignored it.
When the Low Countries and France fell the US further flat out told Germany that under the Monroe Doctrine that they would not allow an Axis occupation of any Dutch, French, or Danish territory in the Americas by a Senate resoultion passed on 6/17/40. Two days later a the second meeting of Foreign Ministers of the American Republic at Havana the Act of Havana was unanimously approved establishing a "collective trusteeship" of colonial territories threatened by "Nazi occupation."
Two months later on August 18, the US entered an agreement with Canada setting up the Permanent Joint Board of Defense to study "the north half of the Western Hemisphere" defense problems. At the same time Congress was authorizing almost $50 billion in new defense spending and calling for a "50,000 plane air force."

Its pretty clear that right from the start of the war that the US had absolutely no intention of letting Germany get so much as a foothold anywhere in the Americas. Had Germany gotten one or another nation to throw in with them in South America it would have been to that nation's great disadvantage. Given the political and economic realities of the Americas it is hard to see how any nation, however sympathic they might be to the German cause, would have been willing in the existing atmosphere to actually openly side with Germany.