Luke, a warm welcome to the board, and thanks for reminding us of the classic translation that is dominant! Like most, all my training was with the BDAG lexicon (though we were not to take it as inspired, but to do the only thing they could do: study the ancient Greek! For no dictionary can settle debates over an author’s meaning apart from wrestling with his usage in context, and Arndt and Gingrich’s discussion is much more complex than you quoted).
I am less sure that I follow your reasoning as to why all other meanings are “impossible.” On your rationale 1: In every account of events considered “eschatological,” are you saying that they are always of “unending duration”? On your second argument, when you say that all judgment passages have “equally viable outcomes,” do you mean that they all are explicitly of unending duration? How would it help either side’s case if “viable” simply meant that given meanings are merelypossible?
My impression is that: (1)many prophetic and eschatological events are explicitly said to be limited in duration. (2) Applying the modifier “final” to judgment passages involves inserting a word there that does not exist in the Biblical text. Whereas many times when the time-frame of a judgment is expressly delineated, it is plainly stated to be limited in duration (or even conditional).
If you nonetheless perceive that what is implied in such passages makes it “impossible” to think there could be any other meaning than one of the human perceptions (one of A & G’s), perhaps you could present examples, and spell out more specifically how you come to that conclusion. It seems to me that the pattern I perceive described in paragraph two would actually incline one against assuming that BDAG’s definition would apply in Matthew 25.