You have it backward. Free Will salvation believers would try to say "they were not his sheep because they did not believe". But Jesus reverses this. He says "they do not believe because they are not his sheep that he died for"
How do I know I'm saved? Because I believe. Had Christ not died for me, I would not believe in any true sense.
Those who choose to believe do not believe or they would not need to force their flesh into a mindset of believing.
I see we have an interpretational difference.
To read the passage the way your stated theology teaches would need it to say,
Ye are not my sheep, that is why ye believe not
instead it says,
But ye believe not, because ye are not of my sheep, as I said unto you. My sheep hear my voice, and I know them, and they follow me: And I give unto them eternal life; and they shall never perish, neither shall any man pluck them out of my hand.
John 10:26-28
I see why you read it as you do. And I acknowledge that reading it thus is one possible way to understand its meaning. Yet the passage has the wording that it does and that wording implies that lack of belief is the sign of exclusion from the sheepfold. Just as following Christ is the sign of inclusion in the sheepfold. The question that you're seeking to answer is "why would they not believe?" and your answer appears to be "because they were created as non-sheep". But pay close attention to the way the story is told by saint John:
And it was at Jerusalem the feast of the dedication, and it was winter. And Jesus walked in the temple in Solomon's porch. Then came the Jews round about him, and said unto him, How long dost thou make us to doubt? If thou be the Christ, tell us plainly. Jesus answered them, I told you, and ye believed not: the works that I do in my Father's name, they bear witness of me. But ye believe not, because ye are not of my sheep, as I said unto you. My sheep hear my voice, and I know them, and they follow me: And I give unto them eternal life; and they shall never perish, neither shall any man pluck them out of my hand. My Father, which gave them me, is greater than all; and no man is able to pluck them out of my Father's hand. I and my Father are one. Then the Jews took up stones again to stone him. Jesus answered them, Many good works have I shewed you from my Father; for which of those works do ye stone me? The Jews answered him, saying, For a good work we stone thee not; but for blasphemy; and because that thou, being a man, makest thyself God. Jesus answered them, Is it not written in your law, I said, Ye are gods? If he called them gods, unto whom the word of God came, and the scripture cannot be broken; Say ye of him, whom the Father hath sanctified, and sent into the world, Thou blasphemest; because I said, I am the Son of God? If I do not the works of my Father, believe me not. But if I do, though ye believe not me, believe the works: that ye may know, and believe, that the Father is in me, and I in him. Therefore they sought again to take him: but he escaped out of their hand, And went away again beyond Jordan into the place where John at first baptized; and there he abode. And many resorted unto him, and said, John did no miracle: but all things that John spake of this man were true. And many believed on him there.
John 10:22-42
Albert Barnes, a Presbyterian pastor and a Calvinist, writes:
Into the sheepfold - The sheepfold was an inclosure made in fields where the sheep were collected by night to defend them from robbers, wolves, etc. It was not commonly covered, as the seasons in Judea were mild. By the figure here we are to understand the Jewish people, or the church of God, which is often likened to a flock, Ezek. 34:1-19; Jer_23:1-4; Zec_13:1-9. By the door, here, is meant the Lord Jesus Christ, Joh_10:7, Joh_10:9. He is “the way, the truth, and the life,” Joh_14:6. And, as the only proper way of entering the fold was by the door, so the only way of entering the church of God is by believing on him and obeying his commandments. The particular application of this place, however, is to religious teachers, who cannot enter properly on the duties of teaching and guarding the flock except by the Lord Jesus that is, in the way which he has appointed. The Pharisees claimed to be pastors, but not under his appointment. They entered some other way. The true pastors of the church are those who enter by the influences of the Spirit of Jesus, and in the manner which he has appointed.
Some other way - Either at a window or over the wall.
A thief - One who silently and secretly takes away the property of another.
A robber - One who does it by violence or bloodshed. Jesus here designates those pastors or ministers of religion who are influenced not by love to him, but who seek the office from ambition, or the love of power, or wealth, or ease; who come, not to promote the welfare of the church, but to promote their own interests. Alas! in all churches there have been many - many who for no better ends have sought the pastoral office. To all such Jesus gives the names of thieves and robbers.
This is much more the intended purpose of the passage, much more what saint Joh intends to teach, than is the idea you've attributed to the words "because ye are not of my sheep".
If you would "rightly divide the word of truth" then pay heed to its intended lessons and resist the desire to prove a doctrine. Many make shipwreck of their faith by proving doctrines.
Of these things put them in remembrance, charging them before the Lord that they strive not about words to no profit, but to the subverting of the hearers. Study to shew thyself approved unto God, a workman that needeth not to be ashamed, rightly dividing the word of truth. But shun profane and vain babblings: for they will increase unto more ungodliness. And their word will eat as doth a canker: of whom is Hymenaeus and Philetus; Who concerning the truth have erred, saying that the resurrection is past already; and overthrow the faith of some.
2 Timothy 2:14-18
