First, translating the verb “paradidomi” as “betray” is anachronistic for texts written in the first century. See William Klassen, “Judas: Betrayer or Friend of Jesus,” Chapter 3 The Act of Judas: The Traditional Point of View.
Accordingly, John 13:2 is more accurately translated, “. . . the devil having now put into the heart of Judas Iscariot, Simon’s son, to hand him (Jesus) over.” John 13:2 suggests an act of treachery only because of the involvement of the devil, not because of either the denotation or the connotation of the verb “paradidomi.”
John 13:1-4 is the preamble to the feet washing.
Verse 2 is the preamble’s clue to the need for the feet washing, which need is not addressed further until verse 18, where Jesus adopts the psalmist’s depiction of the trespass against him as an action of the foot: “[he] has lifted up his heel against me.”
We learn about Judas’ trespass against Jesus only from the synoptic Gospels: Judas went and made a covenant to hand Jesus over to the chief priests. (Mark 14:10,11)
We also learn there that Judas was still looking for an opportunity to hand him over when Jesus arose from supper and girded himself with a towel. (John 13:4).
The will of the Father of Jesus was that Jesus not raise Judas up again until the last day, the day that began at sunset during the last supper. (John 6:39)
In addition to raising Judas up again, Jesus also needed to respond to Judas’ trespass against him by telling him his fault so that he could hear him, in order to gain his brother Judas--all the while keeping his response to Judas between him and Judas alone. (Matthew 18:15)
Only Judas understood Jesus’ words, “You are clean, but not all.” (John 13:10) He was at that moment still looking for an opportunity to fulfill his obligation under the diabolical covenant he made.
Only Judas understood Jesus' words, “[he] has lifted up his heel against me.” Jesus’ words clearly depicted Judas’ making of the covenant to hand him over.
Truly, Jesus kept his response to Judas’ trespass against him between him and Judas alone.
Judas heard him, and Jesus gained his brother. (Matthew 18:15)
The effect of the washing by Jesus was to cleanse the one who was not clean. In other words, Jesus cast the devil out of Judas which put it into his heart to hand Jesus over. (John 12:31) Else, what God has cleansed is unclean. (John 5:19; Acts 10:15)
As a result of the cleansing by Jesus, Judas lost his desire to fulfill his obligation under the diabolical covenant he made.
Also as a result, Jesus had to choose and send an apostle to serve as guide for those who would take him. In other words, Jesus had to choose an apostle to eat his food with him. (John 4:34) What should not be a surprise, he chose the one who had already made arrangements to hand him over; he chose the one who had lifted up his heel against him. (John 13:18)
In effect, Jesus converted the diabolical covenant Judas made to deliver him into the cross which Judas had to take up in order to follow him. (Mark 8:34)
Jesus had staged the feet washing as a backdrop against which he could respond successfully to the needs of Judas. Against that backdrop, Jesus cast the devil out of him and gained his brother, Judas.
In anticipation of performing that special act of love for Judas, the feet washing, Jesus seated Judas to his right, the place of honor. Therefore, it was Judas, the newly gained brother of Jesus, who lay on his bosom and, at the beckoning of Simon Peter, asked, “Lord, who is it?” (John 13:25)
What the sop did at John 13:26 was confirm for Judas that Jesus chose him to hand him over.
What Satan did at John 13:27 was make Judas as adamantly opposed to the handing over of Jesus as Simon Peter. (Mark 8:31-33; John 18:10)
Nevertheless, Judas denied himself, took up his cross, and followed Jesus--the first of the twelve to do so. (Matthew 18:24; 20:16)
You see, it is not Judas who is possessed by Satan at John 13:27, but it is those who assume Satan at John 13:27 induced Judas to fulfill his obligation under the diabolical covenant he made who are possessed by Satan. (Matthew 7:1,2) It is they who savor not the things that be of God, but those that be of men. (Mark 8:31-33)
The sop symbolizes the work God gave Jesus to finish. (John 4:34)
Judas accepting the sop symbolizes his agreement to cooperate with Jesus as Jesus finishes his Father’s work.