As you say, matrix mechanics (or the Heisenberg formulation) is equivalent to the Schrodinger formulation, so it has exactly the same range of interpretations as the Schrodinger formulation.

If you want a concrete example of an experiment that would distinguish between MWI and Copenhagen, here it is:

Prepare an electron so that its z-spin state is the superposition |up> + |down> (I'm dropping the coefficients for ease of typing). Have a research assistant enter an appropriately isolated chamber with the electron and measure its z spin. If Copenhagen is correct, this will lead to the collapse of the superposition, and the electron's state will now be either |up> or |down>. If MWI is correct, the electron's state will become entangled with your research assistant's state, and the entire contents of the chamber will now be in one big superposition from your perspective.

Now have your research assistant record the state she measures by preparing another electron in that quantum state. So if she measures |up> she prepares the other electron in the state |up>. Again, if Copenhagen is correct, this new electron's state is either |up> or |down>, whereas if MWI is correct, its state is in an entangled superposition with the original electron and the research assistant. Call this entangled state predicted by MWI psi.

Now you (from outside the chamber) directly measure the difference between the x-spin (not the z-spin) of electron 2 (the one prepared by your assistant) and the x-spin of electron 1. I can't tell you off the top of my head how to operationalize this measurement, but the fact remains that it is a bona fide observable. If you do the math, it turns out that the entangled state psi is an eigenstate of this observable, with eigenvalue zero. So if MWI is right, whenever I make this measurement I should get the result zero. On the other hand, neither of the states predicted by Copenhagen are eigenstates of this observable, so if Copenhagen is right, if I keep repeating the experiment I will get a distribution of different results.

tl;dr: Basically, all I've done here is take advantage of the fact that there are observables that can distinguish between mixtures and superpositions by detecting interference effects.

Of course, in order for this experiment to be feasible, you need to make sure that the system consisting of the two electrons and the assistant doesn't decohere until you make your measurement. With current technology, we're not even close to making this happen, but that is a problem with the feasibility of the experiment, not its bare possibility.