The main alternative paradigm to dark matter is modified gravity (above all, MOND, Modified Newtonian Dynamics). Modified gravity theories usually involve a change at the classical level, but it has been suggested that the galactic rotation curves, etc., might be due to some specifically quantum effect. In quantum field theory, a classical force field generated by an object is understood in terms of virtual particles emitted by that object, the quantum details can cause the field to exhibit properties different from the classical approximation, and maybe virtual gravitons do something funny when they are summed on galactic scales.

If you could explain such quantum modifications to gravity on astronomical scales, as a result of interaction between Everett worlds, then you could have something like your MWI theory of dark matter. The real problem with that, is in finding a coherent causal account of inter-world interaction.

ETA By the way, I suspect that your idea of how MWI gravitational leakage would work, uses paradigms from the wrong multiverse "level". In string theory there is the idea of membranes in hyperspace, each with its own matter attached, and with gravity being the one force that can cross hyperspace. So what we can see in space might be just one brane, and the dark matter could be matter on other nearby branes. I suspect that you're thinking of the MWI multiverse in similar terms - a stack of worlds, with some gravitational interaction between neighbors.

But in QM, quantum interference involves whole configurations at once. Suppose you have two entangled particles x and y, that are far apart in space. Schematically, their combined wavefunction will be x1 y1 + x2 y2 + x3 y3 + ... where the x's and y's are different possible wavefronts for the individual particles, and xi yi is a formal product of two such individual wavefunctions. The point of nonlocality is that the combined wavefunction of the two particles is simultaneously susceptible to changes at the x end and to changes at the y end. It's a nonlocally defined entity whose manifestation as x and y can't be reproduced by influences that travel through the space between x and y - that's the lesson of Bell's theorem.

So the brane-like scenario of locally propagated gravity leaking between worlds that are neighbors in a hyperspace, doesn't capture the nonlocality of quantum effects. What would be needed, is a type of Bohmian many-worlds theory, in which the nonlocal quantum force of Bohm (which allows Bohm to reproduce QM within a neo-classical ontology of determinism and definite trajectories), comes from some sort of nonlocal inter-world interaction. MWIers really ought to be investigating this sort of model, especially modified versions of it that approach locality.