First, let me say beautifully clear explanation of what MWI is and especially what questions it needs to answer.

Except, a peak can have a spread in configuration space. A single peak can be more like a "ridge" stretching between configurations which are classically inconsistent. This already poses problems of interpretation, as does the lack of clear boundaries to a peak... Are we going to say that a world consists of any portion of the wavefunction centered on a peak - a local maximum - and bounded by regions where the gradient is flat??

I don't think this is any more unreasonable than talking about firing two separate localized wave-packets at each other and watching them interfere, even if we don't have a specific fixed idea of what in full generality counts as a "wave-packet". Typically, of course, for linear wave equations we'd use Gaussians as models, but I don't think that's more than a mathematically convenient exemplar. For non-linear models, (e.g. KdV) we have soliton solutions that have rather different properties, such as being self-focusing, rather than spreading out. I guess I don't see why it matters whether you have an exact definition for "world" or not -- so long as you can plausibly exhibit them. The question in my mind is whether evolution on configuration space preserves wave-packet localization, or under what conditions they could develop. I find it hard to even formalize this, but given that we have a linear wave-equation, I would tend to doubt they do.

e.g. to do with relativity

Of course relativity will be an issue. QM is not Einsteinian relativistic, only Galilean (relabeling phases properly gives a Galilean boost), and that's baked into the standard operators and evolution.