I think that there is some disagreement over the exact order of events. I did a Google survey (because all the scriptures just confuse me...), and it seems there are opinions both ways within many different religions. I wouldn't consider that a stupid mistake to hold an opinion held by many people... Which is correct, I don't know. Personally, I think the accounts are flawed because the events never happened.
Yes, there are disagreements among scholars and other religions on whether Judas was there or not. One notion is that Luke's account was not written in chronological order.
But - the WTS claims to accept the Bible as the inerrant word of God, the final authority in doctrine. If that is the case and we read that Judas did not leave until after, then he didn't leave until after...Period.
Unfortunately, the WT, thus with the JWs in tow, will trot out findings by scholars who actually support the notion that Judas left before. They will adhere to that notion that the account is not in chronological order in order to bolster their need for the emblems to be parkaken of only by the faithful.
The WTS conveniently forgets that they accep the Bible as final authority on a subject and allow "worldly" scholars who happen to agree with them to support their belief.
For the WTS, Judas has to have left before or the ritual of the new covenant is worthless. If Jesus knew Judas was going to betray him, yet allowed him to partake, then its meaning loses all significance. That means that anyone today, annointed or not, JW or not, could partake and it wouldn't change anything as to how and what it's about.
The Nisan 14/15 thing is confusing. The WTS has printed an explanation of why they go by Nisan 14 instead of 15. They use a form of calculation that (they claim) is older and thus more accurate than modern Jewish calenders. It's such a minor thing compared to other issues with the Memorial that, to me, it's not worth harping over. Either they're a day off, or the Jews are a day off. What's more troubling is trying to pinpoint the date comparing the different accounts. John's account doesn't line up exactly with Matthew/Marks (or is Lukes?). If you want to argue about a day, start with the discrepency in the scriptures, not secular calenders.