If we are going to accept the gospels as historical documents worthy of critical study and evidence of Jesus' existence then how do we deal with the bits post-resurrection?

Yes, it's quite problematic sorting out fact from fiction in the gospels. This is precisely why there's such a plurality of opinion among scholars as to how much real history can be gleaned from them. But most scholars--even some confessional ones--acknowledge that parts of the gospels owe more to fictionalized literary/theological narrative than history. The post-resurrection accounts are definitely going to be among these (certainly to non-confessional scholars; but maybe to some Christian ones too).

It'd be one thing for a single person to have a visionary experience, or for there to be a crowd that sees some unexplained phenomenon/illusion in the clouds and/or has mass hysteria or whatever (think the Miracle of the Sun). But the gospel accounts of the post-resurrection appearances are structured in a more deliberate, intricate way than this; and they almost certainly owe more to (fictionalized) Graeco-Roman literary tropes of similar postmortem appearances of revered figures (on mobile, but this article will point you in the right direction on that).

How do we critique claims of multiple witnesses seeing an empty tomb or an apparent vision of a risen Jesus?

It's possible that the empty tomb is historical--though it certainly wouldn't automatically mean that the person inside was resurrected from the dead. It might be just as likely, though, that the empty tomb story, too, is simply 'fiction'. There was actually a known literary trope of a body going missing (even from an enclosed space where it'd be other impossible!), and this thought to have been resurrected/immortalized (cf. Endsjø 2008, etc.).

Does the argument that early Christians were 'willing to die' for their belief in the resurrection as fact hold any weight in assessing its truth?

Not at all. I mean, for one, there were stories circulating at the time about Jews enduring all sorts of heinous tortures instead of giving up various ritual practices (etc.). There were all sorts of other fanaticisms in the ancient world, where people were castrating themselves in service of goddesses, etc.

I'd say it's very likely that some of the important early Christians did have 'visionary experiences' of a resurrected Christ. James may be one of those; and we could certainly imagine this to be a factor in strengthening his conviction. James is, of course, also said to have been executed. But his convictions--even in the face of this--do not greatly help us determine the historicity of events (especially claimed supernatural ones).

because it's obvious to all of us that human resurrection is a physical impossibility

Yes, well this is the very question you are posing, isn't it?

This is a point at which one's presuppositions about reality must be acknowledged. For the historian who has decided that supernatural occurences cannot occur, the resurrection cannot be historical. The evidence must be explained on other grounds. For the one who entertains that such explains may be admissable, it still remains to determine whether it is likely, or indeed the best possible explanation. For some in this category, any naturalistic explanation will be preferable to a supernatural one, so even admitting the possibility of supernatural events, one could never admit the probability.

In cases like this do we just have to appeal to common sense over scholarly evidence?

I'm not sure what you mean. Common sense, if it is indeed sense, will be the application of reason to the question. Scholarship, if it holds the name, will be no more than the same, albeit trained in its field and refined through due practice.

Does the argument that early Christians were 'willing to die' for their belief in the resurrection as fact hold any weight in assessing its truth?

Although /u/koine_lingua says "Not at all", I think I must beg to differ. Simply on the fact that willingness to die should at least be weighed as a factor that tells us something about the conviction of the belief held. Many people die for all sorts of beliefs, in this case we might also consider that it appears a number of people died for something that was falsifiable; does it mean that they are credible? "Not at all by itself" I would say, but it should at least be given some consideration.

How Jesus Became God by Bart Ehrman, which came out this year, answers these questions. Its an entertaining read. It goes in depth into the gospels, their accuracy as historical sources, and who wrote them and also explores the likelihood of there being a tomb as the author states that most crucifixion victims would have been unburied and left out for the birds and dogs to dispose of.