Physicists do not directly apply Occam's razor in most circumstances, and we certainly don't do bookkeeping on how many “entities” there are, and your comment illustrates precisely why: how you count is not a given.
Here is something that did happen in classical mechanics: we transitioned from F_i = m a_i to Lagrangians even though they have roughly similar explanatory power. Here is an argument that was not made: “Lagrangians are truer because you don't have to postulate three equations of conservation of momentum and one of conservation of energy, you just have one law of least action.” Nobody even declared a confident end to the tyrrany of Newton's third law as Lagrangians no longer need it.
Furthermore nobody said that classical field theory was “better” per Occam's razor merely because you were no longer bound by the tyrrany of the least action principle and could now consider essentially a world in which F_i = m a_i was not universally true, to be replaced with a philosophical interpretation by some bloke Neverett who declares the fields on-shell “typical” and derives the least-action principle as a statement that “if you find yourself in a typical universe then almost surely your retroactive reconstruction of events satisfies the least action principle.”
No, many worlds interpretation is thriving precisely because it calls physicists attention to the importance of decoherence calculations in the understanding of various physical phenomena. It gives you an idea for how to model measurements that are somehow partial, or being continuously performed. Occam doesn't enter into the discussion in the first place.
In other words, because MWI is obtained by removing a postulate from the usual formulation of QM, I think it's fair to say it's simpler. If, instead, MWI had been obtained by formulating all of QM in some other distinct framework where there was no mention of wave functions, measurements, or the Schrodinger equation etc, and it had one fewer postulate, then yes I would agree that you can't arbitrarily say that it's simpler.
Is that the case with MWI? Is there a constant amount of information at time t and t+1? Note that I see a fundamental equivalence between information and entropy (of the computational sort), and so an exponential growth of computation required to get from t to t+1 is an inescapable theoretical burden.
To put it a different way, MWI seems to reify possibility. But the state of possibility grows exponentially in time, and so the theoretical entities grow exponentially.
Comparing MWI to collapse interpretations, collapse is better regarding this bookkeeping as collapse represents an upper limit to the amount of quantum bookkeeping required. MWI has an exponentially growing unbounded bookkeeping cost.
In QM by contrast we have entanglement, which essentially means that we can't describe one particle separately from all the other particles (if we could, then QM would be just as "easy" to solve as classical mechanics). Instead of 3n functions of time, we instead have a single function of 3n variables (plus time). The complexity of these functions does not scale linearly with n (imagine e.g. a Fourier series in one variable vs one for two variables)
So, you're right that QM is an exponentially harder problem to solve compared to classical mechanics, but this is because of entanglement and has nothing to do with Copenhagen vs MWI.
Well rather than a single world, I think that the perceived identity exists in multiple highly similar and interacting worlds seen as an entity. Just like we have a size in physical space we also have a non-zero "size" in probability "space".
When you have two theories that make the same predictions, but one is strictly simpler than the other, Occam's razor tells us to prefer the simpler one.
In my mind, MWI or something akin to it, is the way to go, and is generally the way I conceptually think about QM.