You're missing my point. To make sense of the Dirac equation, you have to interpret it as a statement about field operators, so locality means (e.g.) that spacelike-separated operators commute. But that's just a statement about expectation values of observables. MWI is supposed to be a comprehensive ontological interpretation, i.e. a theory of what is actually there in reality.

You seem to be saying that configurations (field configurations, particle configurations, it makes no difference for this argument) are what is actually there. But a "configuration" is spatially extended. Therefore, it requires a universal time coordinate. Everett worlds are always defined with respect to a particular time-slicing - a particular set of spacelike hypersurfaces. From a relativistic perspective, it looks as arbitrary as any "objective collapse" theory.