Why do (at least some) Messianic Jews celebrate the birth of Jesus on the 15th of Tishri? In an (attempted) answer to the question of Why is Christmas on December 25th? from the perspective of Messianic Judaism, it was stated that Messianic Jews believe that Jesus (Yehu’shua HaMashiach) was born on the 15th of Tishri (falling in late September or early October in the Western calendar).

What is the justification for this belief? Is it a widespread belief in Messianic Judaism or only held by a minority? Is there any historical evidence that this belief was held prior to the 19th century?
All the Jewish feasts that are described in the book of Leviticus were fulfilled, (or will be fulfilled,) by Christ.

Think not that I am come to destroy  the law, or the prophets:  I am not come to destroy, but to fulfill.  For verily I say unto you, Till heaven and earth pass, one jot or one tittle shall in no wise pass from the law, till all be fulfilled. 
[Matthew 5:17, 18]

Jesus fulfilled the Passover as the Lamb of God without spot or blemish (sinless).

Jesus fulfilled the Feast of Unleavened Bread by His being broken and buried. The Passover seder included the ceremony of taking 3 sheets of unleavened bread; breaking the middle piece and distributing to the guests, the other half of this piece is hidden (buried) somewhere in the house to be found (resurrected) later.

Jesus fulfilled the Feast of the Fisrtfruits of Harvest by rising from the dead.

*But now is Christ risen from the dead, and become the firstfruits of them that slept. 

[1 Corinthians 15:20 KJV]*

Jesus fulfilled the Feast of Tabernacles by being born on that day. The first day of the Feast of Tabernacles is the start of the rainy season in Israel. The Old Testament prophets said that Messiah would drop down like the rain:

*My doctrine [God’s Word] shall drop as the rain,
[Deuteronomy 32:2a]

Then shall we know, if we follow on to know the LORD:  his going forth is prepared as the morning; and He shall come unto us as the rain,  as the latter and former rain unto the earth.  [Hosea 6:3 KJV]*

Jesus was incarnated and born in a tabernacle (Probably built by his carpenter father around a stone manger):

And the Word was made flesh, and tabernacled among us,
[John 1:14a]

Messiah Jesus was born on the first day of the Feast of Tabernacles, as its fulfillment on the 15th day of the 7th month, known as Tishri on the Biblical, Hebrew calendar, as prophesied and promised by God through many Old Testament prophets. 

I would be interested to hear of any Biblical prophecies that anyone has found that say that Messiah will come in the dead of winter, or of any feasts in winter that He could be the fulfillment of. Some have suggested Hanukka, but that occured after the canon of Old Testament scripture was complete; only 168 years before Jesus' birth, and was never part of the Law. The Levitical Feasts had been in effect for over 1500 years. Jesus didn't accept "It's just tradition," as a legitimate reason for belief:

And he said unto them, Full well ye reject the commandment of God, that ye may keep your own tradition. [Mark 7:9 KJV]

Neither should you. Don't just take my word for it; study for yourself to find the truth.

Appended in regards to the following questions: 
"Is it a widespread belief in Messianic Judaism or only held by a minority? Is there any historical evidence that this belief was held prior to the 19th century?" 

The rabbis that I have had discussions with when regarding when they got started all claim to be a continuation of the earliest founding of the church, the body of Christ:

When Peter gave the sermon on Pentecost, all 3000 souls that were baptized were Messianic Jews: that is, Jews who received Jesus as their Messiah:

And there were dwelling at Jerusalem Jews, devout men, out of every nation under heaven. ... Then Peter said unto them, Repent, and be baptized every one of you in the name of Jesus Christ for the remission of sins, and ye shall receive the gift of the Holy Ghost. For the promise is unto you, and to your children, and to all that are afar off, even as many as the Lord our God shall call. And with many other words did he testify and exhort, saying, Save yourselves from this untoward generation. Then they that gladly received his word were baptized: and the same day there were added unto them about three thousand souls. [Acts 2:5, 38-41 KJV]

There were 1000’s of Messianic Jews in Jerusalem who believed, which also kept the Law, which were in the church under James,.

And when we were come to Jerusalem, the brethren received us gladly. And the day following Paul went in with us unto James; and all the elders were present. And when he had saluted them, he declared particularly what things God had wrought among the Gentiles by his ministry. And when they heard it, they glorified the Lord, and said unto him, Thou seest, brother, how many thousands of Jews there are which believe; and they are all zealous of the law: [Acts 21:17-20 KJV]

Peter even felt that he had to act like a Jew when James sent some of the Jewish brethren to him:

For before that certain came from James, he did eat with the Gentiles: but when they were come, he withdrew and separated himself, fearing them which were of the circumcision. [Galatians 2:12 KJV]

After the stoning of Stephen, the persecution of the Messianic Jews became so great that they were scattered abroad to many countries:

And Saul was consenting unto his death. And at that time there was a great persecution against the church which was at Jerusalem; and they were all scattered abroad throughout the regions of Judaea and Samaria, except the apostles. ... Therefore they that were scattered abroad went every where preaching the word. [Acts 8:1, 4 KJV]

But they used this dispersion to reach even more people with the good news of their messiah, but they still preached to other Jews only:

Now they which were scattered abroad upon the persecution that arose about Stephen travelled as far as Phenice, and Cyprus, and Antioch, preaching the word to none but unto the Jews only. [Acts 11:19 KJV]

The book of James was addressed to the dispersed Messianic Jews:

James, a servant of God and of the Lord Jesus Christ, to the twelve tribes which are scattered abroad, greeting. [James 1:1 KJV]

It is said that the book of 1st Peter was also addressed to the scattered Messianic Jews, but it seems a little odd to call them strangers:

Peter, an apostle of Jesus Christ, to the strangers scattered throughout Pontus, Galatia, Cappadocia, Asia, and Bithynia, [1 Peter 1:1 KJV]

You can’t get much farther back than that. All the Messianic rabbis and other teachers that I have spoken to all believe that the Feast of Tabernacles is the fulfillment of the Messiah’s birth, they all celebrate The Passover on the same day that it is celebrated in Jerusalem. And many other teachings that come right out of the Bible. These were all in Orange County, California, there may be other beliefs in other places.