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Are you saying that the second person of the Trinity only became the Son of God at his incarnation. Apart from the incarnation he was still God, but not the Son, just the second Person?

Yes, that is exactly what I am saying.

The Bible tells us in the beginning, He was the Word, not the Son.

To say that the Son was 'eternally begotten' would make one member of the Trinity coming from another and, chronologically coming after another.

Your problem is using personal definitions in a philosophically inexact manner. Besides, that he was referred to by John as the Word, doesn't mean that his sonship was something that started only upon the moment of incarnation. Furthermore, there is no such distinction made in the New Testament to this effect. If anything, the emphasis in John on the preexistence and deity of Jesus was to counter claims of modalism; just as the emphasis on the Word's actual presence in a tangible human body was to counter claims that God only appeared to be in a physical body. Probably more than 98% of the time Jesus referred to himself as "the son of man," the nature he took on when he, the son of God, was begotten. It's better to stick with scripture such as the following:

...the gospel he promised beforehand through his prophets in the Holy Scriptures regarding his Son, who as to his human nature was a descendant of David, and who through the Spirit of holiness was declared with power to be the Son of God by his resurrection from the dead: Jesus Christ our Lord.--Romans 1:4

Note that his being the son of God in the above passage is a nature or relationship that is distinct from his human birth. His human birth resulted in his being a son of David, not the son of God. His resurrection from the dead declared him as the Son of God.

"For what the law was powerless to do in that it was weakened by the sinful nature, God did by sending his own Son in the likeness of sinful man to be a sin offering."

He was God's son who was sent; he didn't become a son by being born as a human.

"But when the time had fully come, God sent his Son, born of a woman, born under law..." Galatians 4:4

Again, the sonship and the sending is prior to the birth as a human. Also this same distinction is found in the descriptions Jesus uses of himself in the Gospels as the Son of Man versus the Son of God.

"This is how God showed his love among us: He sent his one and only Son into the world that we might live through him...--I John 4:9"

Again the sonship and the sending was prior to and instrumental to his appearance in the world as a human being.

"For God so loved the world that he gave his one and only Son, that whoever believes in him shall not perish but have eternal life. For God did not send his Son into the world to condemn the world, but to save the world through him.--John 3:16&17

Again the sonship is prior to the giving and the sending into the world.

"Yes, Lord," she told him, "I believe that you are the Christ, the Son of God, who was to come into the world."--John 11:27

The belief was that Christ was the son of God prior to coming into the world not because of coming into the world. This is seen in what Jesus said to his disciples shortly before being arrested and betrayed:

"I came from the Father and entered the world; now I am leaving the world and going back to the Father."--John 16:28

Note that he didn't say, "I came from God and, upon entering the world, became his Son and he became a Father; now I am leaving the world where I became his Son and going back to him who formerly was not the Father but, by my act of being born through the incarnation, became the Father.