Answering my own question based on Oppenheim's lecture[1]: In his theory, the stochastic interaction between quantum and classical states necessarily causes the wave function to collapse, so it is not compatible with many-worlds. Being able to measure the gravity of other quantum worlds is something predicted by semiclassical gravity, which Oppenheim calls "complete nonsense".

[1] https://www.youtube.com/watch?v=sde7k3jJp5E