Why did the Romans have Judas betray Jesus? Specifically, how did that benefit the case they were building against him?
I'm likely going to get some details wrong as I'm not a student of either the Bible nor Rome.
His "betrayal with a kiss" happens when he tells the Roman guard to arrest Jesus. Were they that likely to arrest the wrong man in a scene out of Life of Brian? Jesus says "Have you come out as against a robber, with swords and clubs to seize me? Day after day I sat teaching in the temple area, yet you did not arrest me."
It doesn't seem like Judas proves the case of blasphemy either. Jesus seems to state that he's the son of God prompting the high priest to say "He has blasphemed! What further need have we of witnesses? You have now heard the blasphemy"
My guess is that it's more of a social function. Just as the Romans were afraid to arrest him during a religious festival (they said, “Not during the festival, that there may not be a riot among the people.”), having an apostle betray him might be evidence that he's not blessed if his closest followers would betray him.
It's also interesting that Judas "would be better not being born" while Peter was sanctified and apparently forgiven by Jesus despite denying he knows him three times. Though there's a far cry between falsely accusing someone for money and denying you know them to save your own skin.
I think you’ve misread some basic details in the Gospels. I recommend at least reading through Matthew 26-27 and John 18-19. The accounts we have do not have Judas betraying Jesus in relation to Roman Law, but in relation to the political and religious dealings of the power-players in Jerusalem, namely the Pharisees and the Priestly groups and Sadducees. Judas arranges to lead the soldiers (probably not Roman soldiers, but armed temple guards) to intercept Jesus at a time when his arrest will not cause a commotion.
The kiss is a standard greeting but it is used to identiy and single out Jesus. While Jesus was teaching in public, not everyone would recognise him by sight. In a dimly lit garden setting, in which the possibility of a commotion might involve Jesus slipping away, the kiss might work to ‘mark’ their primary target.
As for the play between Judas’ betrayal and Peter’s, the way the narrative is constructed is designed to highlight that Judas feels remorse, and then kills himself, whereas Peter experiences repentance, and thus restoration. However you understand the historical underpinning, it is (also) a literary construct in which two responses to betraying Jesus end with radically different outcomes, which invites the reader to a figurative interpretation.
To circle around to your first question, no aspect of Roman Law requires Judas to betray Jesus. Rather, it appears to function in the early stages of Jesus arrest and trial in front of Judean authorities.
I redacted this comment, since it was meant to be an answer to a different post.