I tried searching for sources that directly answer this, but haven't managed to find any, which is why the following is a guess. The Yaavetz, as @JoelK commented, likely based his knowledge on Sefer Yehudit, also known as the Book of Judith. This is likely for two reasons:
Sefer Yehudit appears to describe itself as taking place during the Babylonian Exile, though there's a disagreement between scholars as to what sub-era of the exile it was referring to (for a summary of several different views, see Prof. Yehoshua Grintz's introduction to his edition of the book).
The Yaavetz seems to have had intimate knowledge of the Christian canon, which included Yehudit, among other books. One proof of this is right there in his answer, where he mentions that in the Book of the Wars of Chashmonaim, which is most likely either Maccabees I or Maccabees II (and most probable Maccabees I1), women are not described as key factors in the causation of the miracle of Chanukah. Maccabees during the Yaavetz's time had not yet been translated into Hebrew, and it was also part of the Catholic canon. Another piece of evidence is from a letter he sent to the Vaad Arba Aratzot in 17572 in which he references specific chapters in Luke and Matthew (his seemingly respectful mannerism actually hid deep sarcasm, but he clearly knew what he was talking about).
For these reasons, it seems very likely that the Yaavetz had read Yehudit and that was how he reached his conclusions about the time of the tale. It should be noted that Sefer Yehudit is also mentioned in Shalshelet Hakabbalah. He seemed to think that the story took place in the early days of the Greeks and Chazal decided to use it as a basis for eating dairy products on Chanukah, i.e., they decided to remember those events together with events that are directly related to the Chanukah story.
1 Because Maccabees II is basically a biography of Yehudah the Maccabee, whilst Maccabees I describes the campaigns of several leaders from the House of Chashmonai, hence, perhaps, "Wars of Chashmonaim" in plural.
2 Pinkas Vaad Arba Aratzot, pp. 421-422, compiled by Yisrael Halperin.