Who is "James?"

[Difflugia]

Two of the twelve apostles were named "James:"

James, son of Zebedee
James, son of Alphaeus

Your later harmonization notwithstanding, Mark, the earliest Gospel, knows three different men named James. There's nothing connecting "James the Less" (Mk 15:40) to either of the other two, even if we allow the reasonable (but hardly indisputable) assumption that the Mary and James referred to in Mark 6:3 are the same "James the Less" and Mary of 15:40.

James of Alphaeus' brothers [siblings] were Apostle Jude (Judas/Thaddeus) and Joseph.
Thus it is indisputable this James-apostle is the "James" in Mt. 13:55/Mk. 6:3 and Ga. 1:19.
(Mt. 10:3, Mk. 3:18, Lk. 6:15-16, Ac. 1:13)

I'm not sure how you're getting from A to "indisputable" B. The only two called "sons of Alphaeus" are James and Levi (Mk 2:14).

The mother of James, Simon, Joseph, and Jude (Judas/Thaddeus) of Alphaeus was also named "Mary," based on the following:

"Mary, mother of James" (Mk. 16:1)
"Mary of James" (Lk. 24:10)
"Mary, mother of James and Joseph" (Mt. 27:56)
"Mary, mother of James the Less and Joseph" (Mk. 15:40)

I hope you realize that this argument actually damages your "indisputable" thesis. According to Catholic doctrine, the mother of Jesus and the mother of James aren't the same Mary. That means that while the two Marys are different, you call it "indisputable" that the same two verses must refer to a single James. Furthermore, you still haven't connected any of the named people to Alphaeus, nor have you established that any Gospel Judas (Iscariot, brother of Jesus, or son of James) was referred to as Thaddeus.

The scriptural verses and testimonials of prominent early Christians above illustrate "James" in Mt. 13:55/Mk. 6:3, "James the brother of the Lord," "Apostle James of Alphaeus," "James the Bishop of Jerusalem," "James the Less," "James the Just," and the author of the Epistle of James were the same, and that he, Simon, Joseph, and Jude (Judas/Thaddeus) were sons of Josephs' brother [sibling], Alphaeus (Cleophas/Clopas), and Mary of Josephs' sister-in-law, Mary of Cleophas/Clopas, and thus Jesus' brothers, as in "kin," specifically, cousins. This disproves the teaching that states they were Jesus' step-brothers and undermines the teaching that states they were half-brothers.

You've established that the texts have been harmonized that way, but not that such is the most likely (let alone "indisputable") reading intended by any (let alone all) of the Gospel authors.

Note: There was no word for "cousin" in Hebrew and Aramaic.

The Greek word ἀνεψιός specifically means "cousin" as opposed to ἀδελφός, "brother." The New Testament was written in Greek.