Is there anything in Christian canon that stipulates a belief in God is required to call oneself a Christian?
Well, if you want to get pedantic, there is nothing stopping you from claiming to be "Christian" regardless of your beliefs. Indeed, this very topic can be hotly contested on this very site with respect to LDSs and JWs.
To actually be Christian means, literally, to follow the teachings of Christ. If I were to trot out everything Christ is recorded to have said, there would be plenty of instances where following Christ's teachings would be... challenging (to say the least) for someone that disbelieves in God. (On that note, see also this question.)
That said, one might summarize the Christian message as follows:
- God Created the world, humans, and you. This Creation was Perfect ("Very Good").
- Humans (Adam and Eve) broke God's command; that is, they sinned.
- God sent his perfect Son to atone for sin... yours, mine, and everyone's, everywhere, everywhen.
- That atonement was accomplished by Christ's death and resurrection.
However, I also want to look at something else you mentioned:
A reason an atheist would want to participate in Christian social frameworks [...] is if they compartmentalize the foundational dogma/mythology, which they find false or contrary to evidende (sic)
I surmise that here you are talking about "science", and particularly the outgrowths of philosophical naturalism (PN) which are Uniformitarianism and Evolutionism (collectively, U&E). Now, PN on its own necessarily denies God's existence; indeed, doing so is in many senses the very definition of PN. Accordingly, while many people may believe in God and Christ while simultaneously accepting the "accepted science" of U&E, these philosophies are fundamentally in contradiction.
Now, it should be obvious that Christ's Purpose is rendered void if there is no Sin. It should be at least somewhat obvious that the notion of Sin makes no sense without Law, and there can be no Law if there is no God. Thus we have our first serious objection to your premise.
What's less obvious is that the entire concept of sin makes no sense if the world was not Very Good prior to the Fall (hence the first point). The important point here is that a Very Good Creation is not compatible with billions of years of death and suffering (i.e. U&E; further reading). Worse, what too many Christians don't realize — because Satan and his anti-Christian minions have worked long and hard to bury this understanding — is that "the evidence" is NOT contradictory to Christianity, including a plain reading of Genesis that includes Creation in ~144 hours (as measured on Earth) approximately 6-10 kya. Rather, there are interpretations of evidence, made from an anti-Christian perspective, which purport — dishonestly, maliciously and falsely — to repudiate the history of the Bible, which comes from God and is reaffirmed by Christ in Mark 10:6.
Moreover... Christ certainly believed in God. (Christ is God, according to Trinitarians, but that's a whole other topic.) Therefore, one who believes in Christ ought also to believe in God. One who believes in God has no need to resort to the extreme lengths UNs to explain Creation. Interpretations of available scientific evidence which are also consistent with biblical history are well known, well studied, and at least as consistent as U&E interpretations.
Accordingly, I must additionally question your supposed need for a "Christian" that does not believe in God. I would commend you (and every other Christian) to start from Christ and seriously examine any competing beliefs (U&E are beliefs; they are not "facts", and they have numerous and significant problems) and whether or not they are really "false or contrary to evidence". Don't just take the PNs word for it; their philosophy is fundamentally opposed to Christ, and vice versa (n.b.
1 Corinthians 1:23, and also 2 Thessalonians 2:10-11).
That all said, I'm tempted to be contrarian and say "sure, why not!". Humans are surprisingly good at doublethink. Plenty of Christians accept Christ's message and the anti-Christian messages of PN/U&E. It may be easier in that case because the contradiction is perhaps less obvious, yet if many people can do that, why not simultaneously believe in Christ and disbelieve in God? I would venture even to say that most Christians (and yes, that includes myself) believe in something that is contrary to God's Word.
If you're doing so consciously and deliberately, that's probably an issue, but I am also not the Judge of Man (and am trying to be mindful of Matthew 7:3-5).