It seems to me that the answer is 'Yes' and 'No' because the institution of the Last Supper is carefully woven into the celebration of the Passover meal and Judas seems to have left during the Passover meal. 

According to Sinclair Ferguson, the Passover celebration had 4 cups of wine. 
- Cup #1 was lifted before the beginning of the meal. Then unleavened bread, butter herbs, stewed fruit, greens and roast lamb was brought in, and the youngest person would ask the traditional question: 'Why do we eat these foods on this night?' - in reply the father would recount the Exodus and then everyone would sing Psalm 113-115.
- Cup #2 was then passed around - and the plate of unleavened bread was lifted up, with the words 'This is the bread of affliction which our fathers ate in the land of Egypt. Let everyone who hungers come and eat; let everyone who is needy come and eat the Passover meal!' - the the head of the household would break off a piece for each person present. The bread was normally eaten in silence, but at the Last Supper Jesus spoke: 'I myself am the Passover Bread' (Ferguson) His life would be given as a substitute for His people!
- Cup #3 after the finish of the meal was blessed and passed around. Psalms 116-118 were then sung. 
- Cup #4 close the meal. 

Again Ferguson asserts that it was as the 3rd cup was about to be drunk that Jesus interjected a further 'new' note. "This cup was the cup of the new covenant sealed in His blood!
Also it seems that Jesus announced that He would not drink the 4th cup. It also appears they went out and sang Psalm 116-118 or they sang the Psalm and went out? 

Mark 14:26 ESV
>And when they had sung a hymn, they went out to the Mount of Olives.

All this to say, Jesus celebrated the Passover meal with the disciples, but He accommodated that meal into the Lord's Supper and instituted it for all His followers 

Luke 22:19–20 ESV
><sup>**19**</sup> And he took bread, and when he had given thanks, he broke it and gave it to them, saying, “This is my body, which is given for you. Do this in remembrance of me.” <sup>**20**</sup> And likewise the cup after they had eaten, saying, “This cup that is poured out for you is the new covenant in my blood.

Judas was clearly there for part of the celebration, and the Lukan account seems to have him there through the 3rd cup. John's detail could suggest an earlier departure -perhaps even at the 1st cup? Matthew and Mark seem to keep Judas there longer than the 1st cup, but not clear how much longer. 

The whole process was much more organic than we make it sound - and if Judas left in the middle - it makes little difference theologically. Jesus knew that Judas would betray Him long before the Supper, long before Judas knew. 

John 6:63–64 ESV
><sup>**63**</sup> It is the Spirit who gives life; the flesh is no help at all. The words that I have spoken to you are spirit and life. <sup>**64**</sup> But there are some of you who do not believe.” (For Jesus knew from the beginning who those were who did not believe, and who it was who would betray him.) 

And yet Jesus allowed Judas into the disciple band. Allowed Judas charge of the money bag etc., Judas was also an 'Apostle' according to the New Testament. 

Also, there appears to be a Manuscript discrepancy with John 13:2 the TR has 
>γενομένου (being ended) 

while the Morphological and Nestle Allen have 
>γινομένου (during) 

I am a TR guy myself, but understand that many will not see John 13:2 as proof that Judas was there through the whole meal.