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.
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.