There are expositors who also think the Darius referred to at Neh. 12:22 is Darius II (Nothus). I wouldn't solely lay this identification at the WTS's door nor be hasty to say they are mistaken this time.
See http://biblecommenter.com/nehemiah/12-22.htm
The NIV Bible Commentary (1994) I have simply says:
"Darius the Persian" was either Darius II (423-404 B.C.) or, less probably, Darius III (335-331 B.C.), the king whose empire Alexander the Great conquered.
The New Jerusalem Study Bible also tags him as Darius II.
Some think Josephus' account was unreliable here. One of the Elephantine Papyrii indicates that a 'Bagoses' lived in Darius II's time. I haven't examined it in detail, but there's enough to show there's more to this question than meets the eye.
And there are different interpretations on how to count the 'weeks of years' to Jesus the Messiah, or who gave the 'word' to rebuild Jerusalem and when.
Just to add: The WTS is provably wrong in dating Artaxerxes I's 20th year to 455 BCE, and is confused on the kings mentioned in Ezra 4-6.