In the Chronological Bible of F. LaGard Smith he harmonize the passages just like that:
Importance of ocassion: Luke 22:14-16
Memorial established: Luke 22:17-20, [Matthew 26:26-29], [Mark 14:22-25]
Jesus clues about the traitor: Luke 22:21, 23
Apostles dispute: Luke 22:24-30
Jesus washes the apostles feet: John 13:1-5
Peter dude: John 13:6-11
Jesus explain why it washes feet: John 13:12-17
Jesus predict the betrayal: John 13:18-21, [Matthew 26:20, 21], [Mark 14:17, 18]
Traitor identification: Matthew 26:22-25, John 13:22-27a, [Mark 14:19-21], [Luke 22:22]
Jesus call Judas to go out: John 13:27b-30
I not revise it, but is a base for begin.