From what I can tell after reviewing the gospels, John tells us exactly when Judas left, which was "but nox" (de nox), meaning just before the night watch which is from 9 pm to midnight. But this was after Jesus had returned from washing the feet of the disciples. The new covenant, though was instituted during the main part of the meal. Judas left IMMEDIATELY after Jesus gave him a morsel to eat. But, again, this was after he had washed the feet of the disciples, at time when Judas was still present, obviously.
Even if we presume Luke is out of chronological order, the "logical" order of events indicates that the heated discussion over who was the greatest stemmed out of discussion about who would betray Jesus. Once that discussion got going, it inspired Jesus to wash the feet of the disciples, showing humility.
Now, excluding entirely, I think it is important to compare John and Matthew. Matthew clearly shows that the new covenant was instituted while the meal was still going on, thus early in the evening. By contrast, John specifically notes that Judas was given the morsel and told to leave "immediately" after that, after Jesus had washed the feet of the disciples, which was just before 9 p.m. a time which logically would be after the main eating of the meal was over. That means Judas was included in partaking of the new covenant.
But why not? Particularly since there was a fulfillment of the one who ate with him and broke bread with him would betray him. This could specifically be a reference to someone who had been part of the new covenant, since the new covenant was symbolized by the eating of bread and drinking of wine.
That having been noted, I don't know how anyone can say that the events in Luke are out of chronological order. the basic order of events, when you combine all the gospels are that Jesus began to eat that meal and almost immediately brought up that someone would betray him, at which point each of them wondered if it was he and asked was it he? Jesus pointed out it was the one who was dipping in the common bowl with him at which point, Judas, in turn, asked if it was him? Then as they continued to eat, Jesus instituted the new covenant, which included Judas. As the meal progressed, the coversation got into who was the greatest among them and the discussion became quite heated. So Jesus took this opportunity to demonstrate to them about humility and began to wash the feet of all the disciples. Now there were 12 of them and some had objections, like Peter. So how long did that take?
After Jesus finished washing the feet of his disciples and reclined again at the table, the issue of who would betray him was raised again. This was just before the night watch at 9 p.m. When John asked Jesus who it was, Jesus then dipped a morsel and then gave it to Judas. This is not the same thing as Jesus and Judas dipping in the common bowl earlier. Then Jesus told Judas to get done more quickly what he had to do. I guess Jesus figured it was already getting late. To IMMEDIATELY after Jesus gave Judas the morsel, he left. John then tells us the specific time. he says in a separate sentence at John 13:30: "It was but night" which means it was just before the night watch, thus just before 9 p.m.
Then it appears after that they sang some karaoke songs (probably something from the Supremes) up until midnight at which time they went into the Garden of Gethseme. :>
Now, granted, I still may be missing it, but I don't see where anyone can claim that Luke has any events out of chronological order. ??
Implying that his account is out of chronological order has to be critically confirmed and aggressively questioned. I don't know how you can do that when the other accounts are not that specific. Nothing I've seen in Luke contradicts anything the other gospels present. So I think this stems from the idea of some not liking the idea that Jesus would include Judas as part of the new covenant when he knew he was the one who would betray him. But maybe it was important that someone who was part of the covenant, that is, someone very close to him and in his inner circle betrayed him.
This may reflect on the references that Satan entered Judas. Thus, the reference that someone who ate bread with him betrayed him reflects back to Satan. You see, Satan and Jesus were in a eunique position. In heaven, Satan and Jesus are represented by the two covering cherubs on the ark of the covenant, which represented Mount Zion. So Satan was in a unique position as far as being close to Jesus is concerned. In fact, in Genesis 3:15, a reference is made to Satan and his rebel angels as "the woman and her seed." Satan is called "the woman" because in a spiritual sense, sharing that unique position on mount Zion with Jesus, Satan was considered Jesus' wife. So on a higher spiritual sense, the betrayal by Satan was most personal. Satan, being the intimate companion and "wife" of Jesus while in heaven is now betraying him. This was prophesied in Genesis where the serpent would bruise Jesus in the heel by having him put to death while a human.
So far from there being any contraindication for including Judas, there seems to be just as many indicators that Judas needed to be included in the new covenant to fulfill these concepts of someone close to Jesus betraying him.