Good exposition but I think that your fourth dimension explanation doesn’t work, although I also think it isn’t necessary. Certain mathematical properties apply to spatial dimensions , for example that directions have no intrinsic meaning, so you can rotate objects arbitrarily within them, and also that if you had a fourth spatial dimension then it would have to be shared between all branches. It does not in itself provide a mechanism for differentiating different branches, since it only offers a single new dimension of distance. So you could then theoretically locate every event in the multiverse in only five dimensions (your four plus time), but you can’t. You can only locate them by navigating the branches. It’s the wrong structure. Those same constraints on dimensions are why the higher dimensions in string theory are considered to be curled up. Otherwise you could rotate objects through them and the world would not look three dimensional at all.
I don’t think the dimension is necessary though because it is sufficient to make it a postulate that decohering branches don’t communicate. That’s just the structure of the world.