It can be determined by finding out when Zachariah was doing his priestly duties in the temple. It was the "course of Abia" Luke 1:5-8. Now Elizabeth got preggers then and 6 months later, Mary got pregnant. 9 months later Jesus was born. Through a long and intricate study, we find when the course of Abia, or "Abija" in some translations, was,( in the OT), and add 15 months you get Tabernacles.
Now that is just facts we can deduce from the Bible.
In the symbolic realm, Tabernacles celebrates the "Light Of the World". A pole has swaddling cloth wrapped at the top dipped in oil and is lit. The priest declares "Behold the Light of the World", as he points to the lit pole in the courtyard. Tabernacles is the only Jewish Feast for the whole world. Jesus came for the whole world. All others are for Jews only. In John 1, we see the Word was in the beginning, the Word was with God and the Word was God, clearly meaning Jesus. In verse 14 we see, "And the Word became flesh and dwelt among us, and we beheld His glory, the glory as of the only begotten of the Father, full of grace and truth." The word "dwelt" there is actually the word "tabernacled", and is the only time in the Bible it is used in that fashion. There are many more reasons to believe Jesus came to us on Tabernacles, but these are the most convincing. Each time a Temple or Tabernacle was built for God to "dwell" with His people, it was dedicated on Tabernacles.
On a side note, you hear many Christians and Jews talking today about rebuilding a new Temple on the Temple Mount. If that temple is built it will be for or by the Antichrist. Revelations tells us that the Second Coming of Christ will be the New Temple building time. Jesus will first re establish the Sabbath on Saturday, and bring with Him the plans for the New Temple.
Alfred Edersheim accepts the traditional date based on the following historical evidence:
At the outset it must be admitted, that absolute certainty is impossible as to the exact date of Christ’s Nativity - the precise year even, and still more the month and the day. But in regard to the year, we possess such data as to invest it with such probability, as almost to amount to certainty. ...6. Lastly, we reach the same goal if we follow the historically somewhat uncertain guidance of the date of the Birth of the Baptist, as furnished in this notice (St. Luke i. 5) of his annunication to his father, that Zacharias officiated in the Temple as on of ‘the course of Abia’ (see here vol. i. p. 135). In Taan. 29 a we have the notice, with which that of Josephus agrees (War vi. 4. 1. 5), that at the time of the destruction of the Temple ‘the course of Jehoiarib,’ which was the first of the priestly courses, was on duty. That was on the 9-10 Ab of the year 823 A.U.C., or the 5th August of the year 70 of our era. If this calculation be correct (of which, however, we cannot feel quite sure), then counting ‘the courses’ of priests backwards, the course of Abia would, in the year 748 A.U.C. (the year before the birth of Christ) have been on duty from the 2nd to the 9th of October. This also would place the birth of Christ in the end of December of the following year (749), taking the expression ‘sixth month’ in St. Luke i. 26, 36, in the sense of the running month (from the 5th to the 6th month, comp. St. Luke i. 24). But we repeat that absolute reliance cannot be placed on such calculations, at least so far as regards month and day. (Comp. here generally Wieseler, Synopse, and his Beiträge.) (Life and Times of Jesus the Messiah, Appendix VII.)
What’s your source on the wrapping of the pole? I like that, but I’ve never seen it before.