It seems clear to me that if conscious memory is predictive of future physical experience, it is drawn from something local to the Everett Branch my consciousness is in.

Whatever makes you think that your consciousness is in only one Everett branch? (And what do you think is happening on all those other branches that look so much like this one but that lack your consciousness?)

Surely the right account of this, conditional on MWI, is not that your consciousness is on a particular branch but that each branch has its own version of your consciousness, and each branch has its own version of your memory, and each branch has its own version of what actually happened, and -- not at all by coincidence -- these match up with one another.

What happens to your consciousness and your memories is much more like splitting than like collapse.

(It sounds as if you think that this ought to mean that you'd have conscious memories in one branch from other branches, but I can't see why. Am I missing something?)