Without addressing the Greek language, there are many who take both sides in interpreting this:
Some say he did eat the Passover with them.
Some say he did not eat the Passover with them.
I am open to whatever makes sense.
Note some of the following details and information, some of which may bear upon the answer:
The wording about his desire might indicate that he really wished he could partake with them. Might this imply that he could not?
I am unaware of any of the accounts indicating that there was cooked lamb at The Last Supper. This may or may not be indicative of whether The Last Supper was an actual Passover meal or something else.
Perhaps he was not to eat of it because he was to become "our Passover sacrificed for us"!
Some have suggested the he was Pescatarian, and might have not desired to eat most Passovers (that included flesh from a lamb), but during this one something incredibly special was to occur.
Perhaps he was expressing his great sentiment to be with them, regardless of the circumstances, or even dietary choices that may have been made by him and/or by certain or all of the Apostles. (I am aware of what it says in Romans 14 and other passages regarding diet.)
The killing of the Passover lamb seems to be unmentioned anywhere in the accounts. Not wishing to build an argument from silence – which I think would be weak here – it is conceivable that the sacrifice had not been made yet.
Some have believed that he was killed or impaled or died at the time that the Passover lamb would have been. (I haven't examined this claim.) Might the timing of his whipping be relevant? ("By his stripes we are healed.") Of his beating? Of the placement of the "crown of thorns"?
Tangential or not: It is conceivable that that year, the Jews had started the year on an incorrect day. In certain traditions, the year began on a day that was related to the sighting of barley ears (or something like that). New Moons were a tricky thing in some traditions too. What if they got it wrong that year?
Note that in the original post, you wrote "eat it", but the Greek word "autos" says "thereof" (in a linked Interlinear) meaning in English "of it". I don't know whether or not this is significant. But I did notice the difference.
Note the significant phrase "between the two evenings", which some source says was when the Passover was to be slain. This is also a phrase (from some source) that some have argued indicates a hint at the darkness that came over the land: Thus, Yahshua of Nazareth could have fulfilled that phrase (technically), regardless of the actual timing in relation to the regular Passover timings.
Passover timing: There has been discussion that there was some disagreement as to when the Passover was to be done or celebrated. There has even been discussion that there was disagreement as to which day was actually the Passover! I don't know all the ins and outs of all this, but perhaps it differed among various groups.... (I may have heard that it differed among the Samaritans?) Today, of course, there are differing traditions as to how and when to do Passover and/or other "Holy Days" (feasts, fast, etc.). On one calendar it may show that in the "Diaspora" (in this usage, meaning outside of Israel), two (2) days may be required for or other-wise kept for a certain event! (but not necessarily for all events).
Because he was to become "our Passover sacrificed for us" – and because he may have been given authority (as the Son of Man may have had authority over Sabbath) – he might have had the authority to institute a different day for Passover! For while the original meal and time spent were significant to the Israelites in the events of the Tenth Plague passing over them (and of part(s) of their Exodus), so too the sacrifice of Yahshua of Nazareth, as he tasted death for all people, was significant to all peoples. Perhaps a NEW Passover was being made – one in which not only could the Jews celebrate fully, but – one in which all nations could partake! (I have never heard this or read this before. It just came to me as I was typing this today.)
Some interesting things to think about?