The simple reply to the conumdrum is that the Monroe Doctrine was never intended to be whatever the opposite of "retroactive" is (prospective?).
There is no independent clear assessment of the status of the Falklands, indeed, of the very existence of "Argentina" as an independent stable state in 1823.
The territory that ultimately became Argentina (and a few other current countries) was in a constant state of civil war or isurrections the entire first half of the 19th Century.
What finally became Argentina didn't even exist when the Monroe Doctrine was established.
Regardless of the national entity, isn’t the presence of the European power the violation? Latin America was gaining its independence from Spain in the first half of the 19th century; I don’t see how that impacts the doctrine.