What did Judas betray, and why did he betray it?

Kiss of Judas (1304–06), fresco by Giotto, Scrovegni Chapel, Padua, Italy

It is simply too trivial to affirm that Judas betrayed Jesus' location. It was all too easy for the Jewish authorities (as well as for the Romans) to keep under watch his whereabouts. Besides, the Gethsemane is a place where "Jesus had met there many times with his disciples" (John 18:2)

It is not true that Jesus openly claimed to be the Messiah. He NEVER claimed it openly, but reserved this claim (or, to be precise, the non-denial of this claim) ONLY to the strict circle of his Apostles.

ONLY when Joseph Caiaphas interrogated him, after Judas' betrayal, on his claim to Messiahship ...

But Jesus was silent. The high priest said to him, “I charge you under oath by the living God, tell us if you are the Christ, the Son of God.” (Matt 26:63)

...  Jesus immediately and openly admitted that he was the Messiah, BUT in a heavenly, eschatological sense:

Jesus said to him, “You have said it yourself. But I tell you, from now on you will see the Son of Man sitting at the right hand of the Power and coming on the clouds of heaven.” (Matt 26:64; cp. Ps 110:1, Dan 7:13)

This, BTW, is Bart Ehrman's conclusion (see The lost Gospel of Judas Iscariot, 2006, chapter 10, What Did Judas Betray and Why Did He Betray It?)

The early disciples tried to make Judas' betrayal more "acceptable" by seeing it inscribed in the Scripture. Not only Zechariah (Zech 11:12-13), BTW, but also Jeremiah (Jer 32:9 - who, though, speaks of "seven ounces of silver"), the latter being explicitly quoted in Matt 27:9-10, although Matthew confuses and conflates the two OT prophets.

The Gospels don't even try to veil the unbelievable cowardice of the disciples, although Peter, almost to redeem himself in advance of his triple denial, even resorts to the sword.

Jesus had already foretold Peter's cowardice and denial, but, in spite of all this, he reaffirmed that this would not affect his choice of Peter as leader of the Apostles ...

₃₁ “Simon, Simon, pay attention! Satan has demanded to have you all, to sift you like wheat, ₃₂ but I have prayed for you, Simon, that your faith may not fail. When you have turned back, strengthen your brothers.” ₃₃ But Peter said to him, “Lord, I am ready to go with you both to prison and to death!” ₃₄ Jesus replied, “I tell you, Peter, the rooster will not crow today until you have denied three times that you know me.” (Luke 22:31-34)

... because Jesus also knew that it was only a temporary moment of confusion and weakness, on Peter's part.

In conclusion Judas betrayed to the Sanhedrin Jesus' secret Messiahship. Why did he do it? My guess (only my guess) is that Judas had serious doubts about Jesus' Messiahship, and wanted to put him in a situation whereby he would prove who he secretly claimed to be. Or succumb ...