The American rule for sentences ending in a quotation is that the period should be inside the closing quotation mark. The rule for parentheticals at the end of a sentence is that, when the parenthetical is not itself the full sentence, the period should go outside the closing parenthesis.
My question is about the interaction of these rules -- the placement of a period when a sentence ends with quotation marks inside a parenthetical that is not, itself, a full sentence. My question therefore differs from this question. It's quite close to this one, but neither the questions nor answers there discuss the American rule and its not clear if anyone considered it.
I'm most interested in the case of the first use of a defined term, but am also wondering if the rule is any different for other parentheticals.
Here are two examples with the first use of a defined term in quotation marks. They are written here with the period inside the closing quotation mark.
Compensation of injured employees is addressed by the Illinois Workers Compensation Act (the "Act.")
Before reunification, Germany was divided into two states, the German Democratic Republic ("DDR") and the Federal Republic of Germany ("BDR.")
And here's an example with a longer quote inside a sentence ending parenthetical:
After the third glass, I began to see things as they really were (which, in the words of Oscar Wilde, is often "the most horrible thing in the world.")
(I realize that the parentheses in the last example could, and probably should, be eliminated, but then it wouldn't be much of an example.)
All of these examples, as written, look wrong to me. But I've been indoctrinated to the American rule, so the alternative also seems wrong.
I suspect that if I owned the Chicago Manual of Style, I'd find the answer there.