Let me throw my hat in the ring on this.
I can tell you that December 25th was the codified day of celebration for Christ's birth by at least 354 CE, seeing as how the Chronographer of 354 listed it as such. There are other minor discussions on the subject in the Ante-Nicean fathers' works. But that's the first "set in stone" example of that date being outright recognized as the day.
From a scriptural and historical point of view, I have to agree with someone else who noted that the courses of the priests rotated. There is a lot of confusion in existing works as to the rotation of the courses. Some think they restart on Nisan 1, while others think a variety of other things.
The courses continue in a steady rotation. A course comes on duty on the sabbath at noon and serves a full week until noon on the following sabbath. The twenty-four courses move in steady rotation from one week to the next. The rotation is not paused or reset at the beginning of the year.
This is very simply proven by the fact that the course of Jehoiarib was on duty on the 10th of Ab, when the temple was burned in 70 CE. According to the Talmud, the same was the case when the temple was previously burned.
From the start of the course of Jehoiarib, 168 days have to pass before Jehoiarib serves again (24 full courses of one week each). Ab is only the fifth month of the Jewish year. Even if the preceding four months were all 30 days each leading up to Ab, plus the ten days of Ab itself before Jehoiarib came on duty, that’s only 130 days. Counting back, that would put Malchijah or Mijamin on duty during the first week of the year.
Additionally, there are the Dead Sea Scrolls, which have lists showing a continual rotation, with different courses on duty during different festivals from the very natural shift.
During the relevant year of Christ's probable conception, Abijah was on duty Jan. 7th - 15th, June 24th - 30th, and Dec 9th - 15th. These dates are back-counted relative to the course of Jehoiarib being on duty on the 10th of Ab in 70 CE.
According to Luke, Mary was visited by Gabriel in the sixth month. While it is automatically assumed that this means Elizabeth's sixth month specifically and exclusively, I do believe there is a double meaning in the statement. In the sixth month, the angel was sent to Mary. That sixth month was also Elizabeth's sixth month. In which case, Elizabeth conceived in Nisan, while Mary was visited in Elul, the sixth month of the year, and the sixth month of Elizabeth's pregnancy.
Allowing for forty weeks of gestation, etc., that would put Christ's birth in the vicinity of June (technically late May to early July).
Some will disagree with the sixth month argument, but that's how I see it. And I do think that many other events line up appropriately under those circumstances. Most especially the baptism. The idea that John was baptizing in cold weather is absurd. He would have suffered hypothermia. The Jordan is cold during the fall and winter.