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Each of the twelve apostles came to Jesus, as it is written, and he did not cast any of them out. One of them chose to betray him. Do you think he had no other choice ?

What do you mean by "Do you think he had no other choice?" There is a difference in physically coming to Jesus, and those who are given by the Father to the Son infallibly. Christ already knew that Judas was not a true believer and was not "of His flock." It is because Judas was not of God that He, ultimately, is God's enemy. Because the world is at war with God, and not because God forces them to be at war with Him.

I meant, "Do you think Judas had no other choice than to betray the LORD ?" Of course, this question also implies "Do you think Judas had an opportunity to be saved and enter into eternal life ?" These questions can be simply answered with "Yes" or "No."

As for the other point about the promise, I understand it as Judas initially came to faith and participated in the ministry of the Apostles. The LORD never cast him out; he chose to leave, chose to betray, and in effect cast himself out.

Jesus' teaching about the bread and wine being his body and blood caused many of his disciples to choose to leave him.
Jesus chose Judas.
No one can come to Jesus unless the Father gives him the grace to do so.
It may have been at this time that Judas, who had initially believed, now chose not to believe (like the other disciples). His lack of persevering faith provided an opportunity for more sin (stealing money from that given for the poor) and led to his choice to betray the Messiah.

Then Jesus said unto them, Verily, verily, I say unto you, Except ye eat the flesh of the Son of man, and drink his blood, ye have no life in you. Whoso eateth my flesh, and drinketh my blood, hath eternal life; and I will raise him up at the last day. For my flesh is meat indeed, and my blood is drink indeed. He that eateth my flesh, and drinketh my blood, dwelleth in me, and I in him. As the living Father hath sent me, and I live by the Father: so he that eateth me, even he shall live by me. This is that bread which came down from heaven: not as your fathers did eat manna, and are dead: he that eateth of this bread shall live for ever. These things said he in the synagogue, as he taught in Capernaum. Many therefore of his disciples, when they had heard this, said, This is an hard saying; who can hear it? When Jesus knew in himself that his disciples murmured at it, he said unto them, Doth this offend you? What and if ye shall see the Son of man ascend up where he was before? It is the spirit that quickeneth; the flesh profiteth nothing: the words that I speak unto you, they are spirit, and they are life. But there are some of you that believe not. For Jesus knew from the beginning who they were that believed not, and who should betray him. And he said, Therefore said I unto you, that no man can come unto me, except it were given unto him of my Father. From that time many of his disciples went back, and walked no more with him. Then said Jesus unto the twelve, Will ye also go away Then Simon Peter answered him, Lord, to whom shall we go? thou hast the words of eternal life. And we believe and are sure that thou art that Christ, the Son of the living God. Jesus answered them, Have not I chosen you twelve, and one of you is a devil? He spake of Judas Iscariot the son of Simon: for he it was that should betray him, being one of the twelve.