I personally don't know much about Barbour's theories, but ideas similar to this already occur in physics.
For instance in the path integral approach one sums over the different paths a particle can take between point A and point B to obtain the the amplitude (which is squared to obtain the probability) for the particle to go from A to B.
Each path is essentially the path the particle would take in a world with different laws of classical mechanics.
You don't assume this though, it follows from other things in QM.
It is a different calculational technique with an unusual picture associated with it.
In essence they aren't really other universes, it's just that one can consider a quantum mechanical process in our quantum mechanical universe as a weighted sum of classical processes from slightly different (and continously different) purely classical universes.
Barbour's idea probably contains a similar idea of "universe".
Edited by Son Goku, : Slight edit and expansion.