I see no justification whatsoever for concluding that gravity must therefore be detectable in the weak-field limit.

Suppose we perform an experiment where, based on the measured spin value, we move some macroscopic object with detectable gravity in opposite directions. In the Newtonian background spacetime approach there is no issue with MWI, as both branches live on the same spacetime. In a full GR case, however, the spacetime itself must decohere into different branches, or else we could detect the interaction between different branches gravitationally (I don't know if this has been tested, but it would be extremely surprising if detected). I am not sure what would the mechanism which splits the spacetime itself be, since all current QM/QFT models are done on a fixed background (ignoring ST and LQG). So presumably this requires Quantum Gravity. Yet the whole thing happens at very low energies, slow speeds and weak spacetime curvatures, so that's why I said that this would have to be a QG effect in the weak-field limit. Of course it would only be "detectable" in a sense that if there is no gravitational interaction between branches, then the spacetime itself must decohere by some QG mechanism.