Jesus' date of birth

Luke 1,5-13
In the time of Herod king of Judea there was a priest named Zechariah, who belonged to the priestly division of Abijah; his wife Elizabeth was also a descendant of Aaron. Both of them were upright in the sight of God, observing all the Lord's commandments and regulations blamelessly. But they had no children, because Elizabeth was barren; and they were both well on in years. Once when Zechariah's division was on duty and he was serving as priest before God, he was chosen by lot, according to the custom of the priesthood, to go into the temple of the Lord and burn incense. And when the time for the burning of incense came, all the assembled worshippers were praying outside. Then an angel of the Lord appeared to him, standing at the right side of the altar of incense. When Zechariah saw him, he was startled and was gripped with fear. But the angel said to him: "Do not be afraid, Zechariah; your prayer 32 has been heard. Your wife Elizabeth will bear you a son, and you are to give him the name John.

Luke 1,23-24
When his time of service was completed, he returned home. After this his wife Elizabeth became pregnant and for five months remained in seclusion.

Luke 1,26-31
In the sixth month, God sent the angel Gabriel to Nazareth, a town in Galilee, to a virgin pledged to be married to a man named Joseph, a descendant of David. The virgin's name was Mary. The angel went to her and said, "Greetings, you who are highly favoured! The Lord is with you." Mary was greatly troubled at his words and wondered what kind of greeting this might be. But the angel said to her, "Do not be afraid, Mary, you have found favour with God. You will be with child and give birth to a son, and you are to give him the name Jesus.

Luke 1,32-36
He will be great and will be called the Son of the Most High. The Lord God will give him the throne of his father David, and he will reign over the house of Jacob for ever; his kingdom will never end."
"How will this be," Mary asked the angel, "since I am a virgin?" The angel answered, "The Holy Spirit will come upon you, and the power of the Most High will overshadow you. So the holy one to be born will be called the Son of God. Even Elizabeth your relative is going to have a child in her old age, and she who was said to be barren is in her sixth month.

1 Chronicles 24,3
With the help of Zadok a descendant of Eleazar and Ahimelech a descendant of Ithamar, David separated them into divisions for their appointed order of ministering.

1 Chronicles 24:7-19
The first lot fell to Jehoiarib, the second to Jedaiah,
the third to Harim, the fourth to Seorim,
the fifth to Malkijah, the sixth to Mijamin,
the seventh to Hakkoz, the eighth to Abijah,
the ninth to Jeshua, the tenth to Shecaniah,
the eleventh to Eliashib, the twelfth to Jakim,
the thirteenth to Huppah, the fourteenth to Jeshebeab,
the fifteenth to Bilgah, the sixteenth to Immer,
the seventeenth to Hezir, the eighteenth to Happizzez,
the nineteenth to Pethahiah, the twentieth to Jehezkel,
the twenty-first to Jakin, the twenty-second to Gamul,
the twenty-third to Delaiah and the twenty-fourth to Maaziah.
This was their appointed order of ministering when they entered the temple of the LORD, according to the regulations prescribed for them by their forefather Aaron, as the LORD, the God of Israel, had commanded him.

The bible quotes above contain the key to the question after Jesus' birthday:
Irrespective of the new-year, the month of Nissan starts the counting of the months, the feast seasons and the priestly duty periods in the temple in Jerusalem. Each priestly division had to serve for one week [2]. During the feasts of pilgrimage all 24 division had to serve in the temple.
This means, the priest Zechariah, a descendant of Abijah, could return to his wife not before the end of the feast of Pentecost.

Event	Priestly division on duty	Month	Week
	1) Jehoiarib	1 Nissan	1
	2) Jedaiah		2
Passover	All 24		3
	3) Harim		4
	4) Seorim	2 Iyar	5
	5) Malkijah		6
	6) Mijamin		7
	7) Hakkoz		8
	8) Abijah	3 Sivan	9
Schavuot	All 24		10
Zechariah gets back home: Johns conception	9) Jeshua		11
	10) Shecaniah		12
		4 Tammuz
		5 Av
		6 Elul
Succoth		7 Tishri
		8 Cheshvan
Elisabeth is in here sixth month with John the Baptist: Mary's conception		9 Kislev	No of days since Mary's conception
		10 Tevet	29
		11 Schewat	59
		12 Adar	88
Passover		1 Nissan	118
		2 Iyar	147
Schavuot		3 Sivan	177
		4 Tammuz	206
		5 Av	236
		6 Elul	265
Jesus' birth! Succoth		7 Tishri	280

As shown above Elisabeth did conceive John the Baptist end of the 3. month (Sivan). 6 month later (Elisabeth was in here sixth month), this means end of the 9. month (Kislev), the angel Gabriel came to Mary and she did conceive.
Jesus was born approximately 280 days later (a normal period of human gestation), which gives the seventh month (Tishri). The feast of Tabernacles is appointed for the days between the 15. - 21. Tishri.
Therefore, Jesus was borne around the feast of Tabernacles. [1]

Of course, this argumentation does not lead to an exact dating of Jesus' birthday. We need to consider even two more ambiguities:

The priestly division of Abijah including Zechariah had to serve in the temple a second time around six month later in the year. This boils down to a second possibility for Jesus' birthday approximately six month later during the month of Nissan, perhaps in the time of the Passover feast.
If a leap year lies in the period of the events mentioned above, which means the respective year would contain a thirteens month of Adar, Jesus' birth would be one month earlier.

Nevertheless, there are more considerations pointing in a similar direction:

For climatic reasons, it is hardly possible, that Jesus was born end of December. Example: It is too cold in Bethlehem end of December to find shepherds with there flocks of sheep out in the country during night time.
Ancient records say that December 25 is the date of conception.[3] The perfect period of human gestation is 280 days. That would make September 29 the date of birth of the Son of God, about the time of the feast of Tabernacle.
The word was made flesh, and dwelt (tabernacled) among us. John 1:14
The feast of Tabernacles fittingly symbolises the dwelling of God among men.[3]

[1]
A similar argumentation uses David Pawson in UNLOCKING THE BIBLE, New Testament Bd 1, page 73

[2]
E.g. "Keil and Delizsch Commentary" or David Stern, "Commentary to the Jewish New Testament"