[We are very much in speculative territory here of course, it is not a given that duplication or upload is possible at all for a human mind]

I would say that both A and B would claim to be X and have the "continuous"[1] experience to be X while agreeing that they are distinct persons. I think that the question of whether A or B, both, or neither is the true X, is not a scientific question, and as a philosophical question, a fairly empty one.

Regarding the challenge, I would expect that the consciousness would be forked: there is a "me" that would awake inside the machine and would be very glad to be alive, while the "me" outside would experience dying. This seems to go against the exclusiveness of the experience of being themselves, but assuming the existence of the magically perfect duplication, both would be valid experiences and again neither could make a claim about being the real me. I don't find this to be a contradiction.

An interesting question would be whether consciousness can be reunited after being forked.

But all of these scenarios have been explored extensively. Are you familiar with the Egan's "Permutation City"?

edit: I was not familiar with the functionalist position, but for the little I read, it seems to me that it is just a variant of the monist position. You could say I'm a functionalist I guess. Also I believe that we are fundamentally in agreement and we might just disagree with definitions of words.

[1] what does "continuous" even mean? Do one have a "continuous" experience of being yourself after a night's sleep? After anesthesia? After a coma?