"Liar,madman or God"; CS Lewis on the divinity of Christ

—I recall that Mark and Q were roughly contemperaneous, and both Luke and Matthew used them as sources.—

Actually, Q is suspected of being much earlier than Mark (though since Q itself is only a speculative inference, the arguements on this is are nceessarily extremely weak and speculative). Q is thought to have been the closest thing we have to possible direct quotes from Jesus, before the gospel accounts of birth, life’s politics, death, ressurection, etc. were even popularized. The people that are thought to have written Q are surmised to have seen Jesus more in the vein of a wandering cynic, like the Greek philosophers (indeed, many of Jesus’s sayings are very similar to Greek sayings).
The opposite spectrum is Paul, who is almost definately the earliest author in the NT, but who didn’t have much of anything to say about Jesus’s explicit words or teachings: far more important to his particular theological endeavors was simply that Jesus had died to save gentiles: convert fleshy law into spiritual law: not what teachings he had imparted to anyone.

—Just because it’s [John] the last written doesn’t make it unreliable. If we look at biographies today, we’ll recognize that often it takes time for an authoritative biography to be produced, because so many of the eyewitness testimonies and much of the writers agenda are affected by the controversies of the day.—

Unfortunately, though, “biographies today” exist in a world utterly foriegn to the writers of the Gospels. “Biographies today” are written with an incredible array of cross-checked information with fairly clear and documented chains of custody, standards for citation, as well as a large, well read, and well interconnected scholarly community that is constantly checking up on new works and subjecting them to factual and theoretical criticism. In our time, we have sources ranging from full life’s written works, a diverse investigative press (in later days with video and audio archival footage to search through), a robust system of governmental record-keeping, etc. Many sources, like diaries and letters, are not initially available at the time of someone’s death, but later become so, adding more insight into the person.
All of this leads to an environment in which research can indeed build upon itself, and in which we might expect to have reasonably good (but far far from perfect) certainty in expecting synthesis of relevant material to get better after a few decades of publishing.

So, applying our contemporary assumptions about modern-day biographies to that of the Gospel period is extremely misguided and potentially misleading.

The writer of the Gospel of John lived in a period when information was highly chaotic and confused, primarily oral and much much sparser as far as primary and even secondary written sources, was often guarded by channels of access and even sometimes hoarded as “secret teachings,” as well as being scattered all over Europe and the Middle East, often without any one area having much knowledge of the resources of any other area (no real central system of organization existing to keep track of any of it as it was produced, often in a DIY fashion, often in secret).
No public, centrally interacting community of research existed for Christian works to debate issues of fact, to separate rumor from fact: instead the environment was far more like an immense and ever shifting array of competing splinter groups each developing their own theologies over the decades in response to the various pressures, conflicts amongst themselves, all of their different times and places and indeed just general progressions of theological speculation. Letters advocating this or that view, claiming this or that special revelation or account were common, but the writer of any particular Gospel would have had access to, at best, only a few of these things, with no feasible way to check out or document most sources and accounts beyond simply finding various written records of them, and almost certainly would have their own particular theological school of thought to begin with. Simply stated, there is no reason to think that a Gospel written later on would have been better researched and more accurate than those that came before, and indeed certainly no reason to suspect that it would be anywhere NEAR as accurate as modern biographies, let alone much more accurate, as would need to be claimed to make sure that nothing could possibly be wrong or misleading.
In fact, we don’t even know if the writer of the Gospel of John even KNEW of all the other Gospels: of the three, it is easily the odd man out, recounting events and theological ideas that the others do not, as well as not recounting some of the events that the other texts treat as being central. Indeed, the Jesus of John seems like a very different being than the one in the earlier Gospels: a far less human creature (which makes sense considering the divergent theology that the Gospel seems to recount).

On top of all of this, we are dealing not with simple accounts of someone’s life and history, but extraordinary claims about miracles, gods, secret wisdom teachings about the deepest aspects of life and values, and in general the very sorts of things that are already the most controversial and easily misunderstood (and misrepresented). We are dealing with persecution and acrimony from the authorities over these materials, the short-life spans of potential sources (and the en masse destruction of many records and people in the 70’s), as well as entire communities whose only real source of information is traveling rumor, often in the form of iterant preacher hired to entertain and inform small groups at a time. For us to expect that any text of a relatively new and minority religious group from this era would be a well-researched and accurate historical account would require exactly what literalists claim: a miracle.

—My main sticking point is the idea that we can claim Jesus was misquoted without providing any evidence for it.—

One doesn’t have to, for the very reason that Lewis’ argument is supposed to recount all the possibilities. It is most certainly possible that Jesus was misquoted (indeed, very very likely at least to some extent, though there’s no way to tell what was and what wasn’t). Lewis doesn’t properly acknowledge this, and it’s clear why not: it alone (just the possibility) destroys the entire inevitable power of the deductive arguement he’s trying to make.