I’m starting a new language in December, and my wife is insisting on a living language, otherwise it would be Latin.
You should push back. Knowledge of Latin is a very useful complement to ancient Greek and vice versa. For one thing, you can’t read the prefaces to many critical editions of Greek texts without knowing Latin, and much other scholarly material that was written before about 1850 but that is still valuable was written in Latin.
More importantly, Latin texts often illuminate Greek texts. Although the flow of influence tends to travel more in the other direction, Latin authors often “imitate” Greek texts (“imitation” is really the wrong word, which is why the term “intertextuality” has come to supplant it) in ways that shed light on the meaning of the Greek or help to explain how the Greek was understood by Greeks and Romans alike in the Roman period and later.
You can get by in ancient Greek or in Latin without knowing the other language, but knowledge of both languages will enhance your engagement with each of them.