First, I would acknowledge that the Gospel do rearrange chronology in order to make theological points. This is widely accepted among people who acknowledge the doctrine of plenary and verbal inspiration for a long time, namely from the beginning of the Church! There's no problem in this, because the goal of the Gospel authors is not only to give us a historical account of what went on, but also its theological significance. I agree with David Scaer for example, that it might be that the 5 great discourses in Matthew (the new Torah) could simply be compendiums of Jesus' general teaching and not full versions of speeches that Jesus necessarily gave all at once. I'm certain we can think of other examples of this. The bottom line though this that at the end of the day, Scripture really is truthful in what Jesus literally taught and did, even if it might be rearranged or presented a particular way for theological accent.
This is why Kloha's solution bothered me. The issue isn't that the Gospel writers rearranged things for theological purpose (which they obvious did!), the issue is whether they were truthful about the facts they present. If we think about Kloha's solution, John was quite literally untruthful when he says that the high priest didn't want to go into meet Pilate in order that he not be defiled so that he might eat the Passover. It doesn't matter if it points to some other great theological truth. It's not a rearrangement of material, it's simply a falsehood. Neither does it impress me when people say "well, it doesn't really need to be true" or "it merely illustrates a theological truth, etc." Indeed, I went to a mainline Protestant seminary where this was the exegetical soup du jour. I know where this goes. In the end, the Bible becomes a series of stories which illustrate general, abstract theological truths, and not a really, literal history that happen "pro me." Ultimately, these truth end up illustrating morality and so we move from a faith of the gospel to one of the law.
Furthermore, it's not that there aren't any other solutions to the problem. There are several that could be offered. As an alternative to Kloha's solution, I would make the following suggests that might solve the problem:
1. D.A. Carson and N. T. Wright have suggested that Jesus simply held Passover a day earlier since he wanted to die at the same time that the Passover lambs were being killed in the Temple. This would make what the Synoptics say true, that is, that the last Supper was a Passover meal (just an intentionally early one)- while not contradicting John's statement that Jesus died on the day that most Jews were eating the Passover.
2. David Scaer points out that the Passover festival was a lengthy affair, with several days of feasting. The term "eating the Passover" in Second Temple Judaism has several connotations, not just the eating of the lamb. On the day after the eating of the lamb, the priest slaughtered several bulls which they would then feasted on. Hence, John may be referring to their participation in this sacrificial feast when he says that they did not want to defile themselves for the "eating of the Passover."
3. Jesus and his disciples may have followed the Qumranic calender, rather than the one used by the Hasmonean Temple aristocracy. The Essences at Qumran had a different calender. That's one of the reasons they thought that the Temple was defiled. All the rituals were being performed on the wrong days and therefore didn't actually work. Impurity was building up to unmanageable levels and so God was just going to have to blow everything up and start all over again.
Anyways, Jesus and his followers began in John's anti-Temple movement (why go to the Temple to get your sins forgiven when you can be baptized? Why go to the Temple when Jesus as the new Temple can forgive your sins?). Jesus' movement was of course a continuation of John's. This follows a typical pattern among Second Temple Jewish messianic and anti-Temple movements. As N. T. Wright points out, in this period, if the leader of a movement died or was killed, his relative took over. John started the movement, Jesus took it over as the one who fulfilled John's eschatological preaching. James and Jesus' relatives took it over after he ascended. Eusebius tells us that Jesus' relatives were still running the Church in Palestine in the early 2nd century.
Any how, some scholars have suggested that John might have spent some time with the Essences at Qumran. This would make sense for a number of reasons. 1. They shared John's antipathy for the Temple. 2. Sociologically, they were from the priestly caste, as was John. 3. John's parent were very old, as Luke notes, and they might have died when he was young, meaning that he might have needed a place to live. 4. The Qumranic community understood themselves as "a voice crying in the wilderness, making a path for the Lord" as did John. 5. The Qumranic community placed special importance on ritual bathing, as did John- though in a somewhat different form (i.e., once, not multiple times. Also of course John's baptism was divinely instituted, whereas the Essences was not).
If John was a member of the Essence sect for a time, he might have maintained a different ritual calender than the Temple aristocracy. This might also have become the practice of Jesus and his followers not because they attached any theological significance to it, but because it had become their habit eating it on that particular day when John was around. This could then solve the problem as to why Jesus ate the Passover a day before the Priests.
In any case, any of these solutions is a better than that offered by Kloha. I worry that his solution ultimately calls into question the infallibility of Scripture.