AnnOMaly:
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
Of the seven commentaries I've considered, the majority support Darius III (shown below in bold). Only two say it must be Darius II (shown in italics, of which only one actually connects Jaddua with Darius), with an additional one saying it could be either Darius II (according to James Ussher) or III. Since JW biblical chronology is loosely based on Ussher's chronology, it's also unsurprising that they'd prefer his conclusion.
Barnes':
Probably Darius Codomannus (336-331 B.C.), the antagonist of Alexander the Great.
Clarke's:
Calmet maintains that this must have been Darius Codomanus, who was defeated by Alexander the Great: but Archbishop Usher understands it of Darius Nothus, in whose reign he thinks Jaddua was born, who was high priest under Darius Codomanus.
Gill's:
thought to be Darius Codomannus, the last king of the Persian monarchy, whom Alexander conquered;
Keil & Delitzsch (but adds that the information is "aphoristic"):
Darius is Darius Nothus, the second Persian monarch of that name; … The statements in Nehemiah 12:22 and Nehemiah 12:23 are aphoristic, and of the nature of supplementary and occasional remarks.
Matthew Henry's:
The succession of high priests during the Persian monarchy, from Jeshua (or Jesus), who was high priest at the time of the restoration, to Jaddua (or Jaddus), who was high priest when Alexander the Great, after the conquest of Tyre, came to Jerusalem, and paid great respect to this Jaddus
Pulpit Commentary:
The "Darius" intended is beyond all doubt Codomannus, the adversary of Alexander the Great, who was contemporary with Jaddua.
Wesley's:
Darius Noehus; and so this Jaddua might be father to him who was in the days of Darius Codomanus, and of Alexander the Great.