[crystalk]

Jesus was born on Sept. 11, which was Rosh Hashanah that year, and the Hallel chant was just beginning as he was born.

That is Gregorian. I believe the Julian is Sept. 2.

The Annunciation of his birth occurred on the high day of Hanukkah, the 24th of Kislev, which was taken by Roman Christians later to be something like a December. Due to the Jewish reckoning the days about 6 hrs earlier than the Romans, this caused the 25th Dec to seem like a good choice, which also fell in line with heathen celebrations of the {visible} reversal of the Sun, its beginning to go back became noticeable.

[Leonine]

"...the Hallel chant was just beginning as he was born."
Is this gleaned from Luke?

[Leonine]

"The Annunciation of his birth occurred on the high day of Hanukkah..."
The Annunciation was, of course, nine months before His birth (Gabriel). I think you are referring to the Presentation. The presentation of any first-born Hebrew male was on the fortieth day after birth. By your dates, this would be sometime in October.