"Take the equation literally" is just a slogan and doesn't provide any details.

What it means is that you let your ontology be dictated by the mathematical structure of the equation. So for instance:

Am I to think of Psi as a wavefunction on a configuration space, or as a vector in a Hilbert space?

It's both -- even when regarded purely as a mathematical object. The set of wavefunctions on a configuration space is (the unit sphere of) a Hilbert space. Specifically, as I understand it, configuration space is a measure space of some sort, and the set of wavefunctions is (the unit sphere in) L^2 of that measure space.

Am I to think of myself as a configuration of particles, a configuration of particles with an amplitude attached, a superposition of configurations each with its own amplitude, or maybe some other thing, like an object in Hilbert space (but what sort of object?) not preferentially associated with any particular basis?

It seems to me that you're a region of configuration space. There's a subset of the measure space that consists of configurations that represent things like "you're in this state", "you're in that state", etc. We can call this subset the "you"-region. (Of course, these states also contain information about the rest of the universe, but the information they contain about you is the reason we're singling them out as a subset.)

And then there's that little issue of deriving the Born probabilities!

To repeat a point made before (possibly by Eliezer himself), this isn't an issue that distinguishes between many-worlds and collapse postulates. With many-worlds, you have to explain the Born probabilities; with collapse interpretations, you have to explain the mysterious collapse process. It seems to me far preferable, all else being equal, to be stuck with the former problem rather than the latter -- because it turns the mystery into an indexical issue ("Why are we in this branch rather than another?") rather than writing it into the laws of the universe.

you absolutely need to [break the wavefunction into worlds] to make contact with empirical reality.

Why is this?

my proposition was that if you look at superpositions of these braidings and sub-braidings, you get localized entities which have ontological boundaries and persistence in time until they enter into a larger braiding; and this means you can after all talk about material parts of a person persisting in time.

Okay, it now occurs to me that I may have been confusing "continuity of substance" (your criterion) with "identity of substance" (which is what Eliezer's argument rules out). That's still more problematic, in my opinion, than a view that allows for uploading and teleportation, but in any event I withdraw the claim that it is challenged by Eliezer's quantum-mechanical argument about particle identity.