Good question. Wrong assumptions lead to wrong conclusions. The rock opera Jesus Christ Superstar wrongly portrays Judas as a puppet damned from all time. A Calvinistic view would say that God predestined Judas to do this from eternity past (decree). An Arminian view would suggest simple foreknowledge had God knowing what Judas would freely do. An alternate view called Open Theism is a more coherent, biblical free will theism (www.opentheism.info).
Jesus choose Judas after a night of prayer to the Father. He was called an apostle and was part of Jesus' inner circle. Early in His ministry, Judas went off track and eventually was filled with Satan, became a betrayer, and a son of perdition/hell. This was not fatalistically fixed nor foreknown in eternity past or during his life. There was risk in choosing him, but he was not a puppet. He could have remained faithful or repented after going off track. The Old Testament does not prophesy him by name at all. Like other verses about Jesus that seem out of context, the Spirit applied some verses to Judas by way of illustration/parallel to an OT situation, NOT predictive prophecy like most assume (if it would have named Judas in prophecy, it might be a different matter). Jesus could have died without a betrayer or someone else would have the illustrative verse applied to them if they were the betrayer. If Judas would not have done this or repented, the New Testament (JW Christian Gk. Scriptures) would be recorded differently. God desired for Judas to remain an apostle and would have forgave him if he would have repented instead of showing remorse leading to suicide. Deterministic views are false and free will ones are true. I actually agree with WT view on God's omniscience even though it is a minority view and even considered heretical in Christian circles (though WT is wrong about omnipresence).
So, the Judas thing was not predictive prophecy nor set in stone (this would be unfair/unjust/unnecessary for a righteous God who can orchestrate without coercing). The problem is with the interpretation, not with the possibility that prophecy can be wrong (many prophecies are also conditional vs unconditional). So, you have a non sequitur to think that Judas had to fulfill this and it would make it wrong if he did not (e.g. a verse about Jesus in Egypt is applied illustratively because it parallels a historical situation, but it is not truly predictive prophecy). Judas is culpable for his choices and did not have to betray Christ (someone else could or no one could, so the verse would not have been applied to Judas after the fact if history unfolded differently).
So, there is a third alternative to your false dichotomy that assumes an unbiblical deterministic view. Someone else could have become a betrayer with the generic verse being applied to them, Judas could have repented, or Judas could have not gone off track at all as God desired/intended. The same is true with Christ's death. If soldier x did not kill him, soldier y could have (not settled before Christ was born, etc.). At some point, it was a point of no return for Judas, but this was not the case when he was first chosen to fulfill a positive vs negative role in history. God does not choose a devil, but he freely became one later on.