Unfortunately, this eclipse was visible only very late that night in Judea and was additionally a minor and only partial eclipse.
He says that on the night of the fast there was a lunar eclipse—the only eclipse mentioned in the entire corpus of his work.
There were no lunar eclipses visible in Judea thereafter until two occurred in the year 1 B.
Of these two, the one on December 29, just two days before the change of eras, gets my vote since it was the one most likely to be seen and remembered.
Second, Josephus writes that Herod reigned for 37 years from the time of his appointment in 40 B.
and 34 years from his conquest of Jerusalem in 37 B.