Maximize/minimize the rate of entropy production. "Branching" is in many ways a metaphor for increasing entanglement entropy.

There are no splits. The world state is and always was a complex-valued distribution, and it continuously evolves. Some parts of these distributions, through non-linearity, entropy and decoherence, can be described as "branches", but they're never really separated and they continue to causally affect each other.

Decoherence means that their average effects on each other are near zero, and entropy means that this is for all practical purposes irreversible, but this should not be confused with having created a new universe. It's all still the same one, merely viewed from narrower points of view.

An interpredtation does not provide any additional mechanics on top of the thing it interprets. Thefore its the same thing to ask of just QM.

In terms of quantum circuitly you wuld probably be looking to something equivalent of a controlled Hadamar gate, which is controlled on when you want branching and controlled off when you don't want branching (you have to start from a state that you consider a low number of worlds state (in respect to the qubit it is working on)).

Sean Carrol atleast in some of the speeches does a bit where he queries a spin meaasurement by phone and take a step to the left or right based on the result. You could condition that on only doing it in the bad worlds (say that its is raining). If there is a base chance of 50% rain (2 worlds) then you go to 50% stand center it is is sunny, 25% stand left it rains and 25% stand right it rains (3 worlds). Note that this is not an effective rain dance but it does increase the number of worlds.

What you might have wanted to ask is how to increase the measure of good worlds in comparison to bad worlds. Part of the magic of quantum computing is that we can get results that are more often useful than just running the calculation in each world separately. But even then we mostly increase "good details", we provide stabilitity to the data and chuck the superpositionness to the noise.

In that vein the approach seems often be the opposite, when things are going good you want to keep everything as is and when things are going bad you roll the dice to get anywhere else than here.

As far as I'm aware(having done some work in the foundations of quantum mechanics/concrete models of branching in chaotic systems) nobody knows, there isn't really a well-developed theory of what the 'branch production rate' of arbitrary physical systems is. Based on what we do know -- broadly speaking, creating more chaotic systems with sensitive dependence on initial conditions seems likely to create more branches. That said, even if we had a satisfying theory, I don't think that creating more branches in more desirable worlds makes sense, since those branches would each have a correspondingly smaller probability, reducing their moral value(see comments here for arguments as to why that's the correct way of assigning moral value)

I'm not an expert, but previous questions about Many-Worlds on this site have received no answers from experts, so I'll answer.

I have listened to the audiobook version of Sean Carroll's Something Deeply Hidden about 5 times.

Our world splits often: many times a second or much more than that. I doubt that at our level of science and technology we can exert any influence at all on the rate of the splitting.

You can't freely decide to do anything if MWI is true. Waving that aside, something like Slider's suggestion could work.