Professor Andy Albrecht has kindly pointed me to one of a series of papers by David Deutsch that begins the proof of Hugh Everett’s Many-World’s Hypothesis:
This paper removes the requirement to have a probabilistic
interpretation of quantum mechanics, instead casting it into a
non-probabilistic decision theory basis.
The main result is that the usual probabilistic formulation can be
replaced by a rational decision maker using classic decision theory.
Furthermore (and most importantly), quantum processes whose outcomes
look stochastic can be replaced by deterministic evolution of states.
Note that this neatly removes the issues around definitions of probability in the MWI.