Alice doesn't really make an consistent impression of discuntiuous identity. That's probably intended judging from the title.

there's the fact that I've come to identify with my theory of identity, and also, at the end of every payday, there's a big, virtuous thrill when I send my check.

So she knows that her other mes feel so, she seems to identify with that. That breaks suspension of disbelief (disalief?).

Regarding the recent secretly secret sensations I'd guess that there are probably people out there that do not have all the kinds of identity everyone takes for granted - in this case the agent self.

See http://en.wikipedia.org/wiki/Psychology_of_self#Parts_of_the_self