My understanding of this kind of thing has evolved over time. I used to be convinced that interpretations were inherently untestable, but now I think that was an oversimplification.
To make the discussion more concrete, let's consider a set of axioms for quantum mechanics:
You could add more to this list (observables are self-adjoint operators, completeness), but these are the main things that are important, and they are also things that everyone agrees on. This is all you need for the most austere versions of MWI, which I'll call MWI-basic.
If you want the Copenhagen interpretation, you need some more axioms:
So from the point of view of this kind of axiomatic development, CI is the same as MWI-basic plus additional axioms. One thing this tells us is that any experiment that disproves MWI-basic must also disprove CI.
It is certainly true that MWI-basic and CI are falsifiable. Any observation that falsifies 1 or 2 falsifies all of QM, and therefore falsifies both MWI-basic and CI.
I think the right way to look at this kind of thing is that CI is an approximation, and the approximation is good when the measuring instrument is macroscopic. When the measuring instrument is mesoscopic, the approximation is not perfect, and this is something that we can see. A nice example is Allahverdyan 2017. They simulate a measurement by a mesoscopic system, and they come up with all kinds of phenomena that actually do happen according to quantum mechanics, but that are not correctly described by CI. For example, there are time scales that emerge from the simulation, whereas (4) says collapse is instantaneous.
There are also more baroque versions of MWI, which we can refer to collectively as MWI-baroque. DeWitt gives a description of what I would call a baroque version of MWI:
This universe is constantly splitting into a stupendous number of branches, all resulting from the measurement-like interactions between its myriads of components. Moreover, every quantum transition taking place on every star, in every galaxy, in every remote corner of the universe is splitting our local world on earth into myriads of copies of itself.
This is also an approximation, and the approximation is not perfect. The approximation is valid if decoherence leads to a set of preferred states that are not "cat states," i.e., not coherent superpositions of different pointer states (like Schrodinger's cat). This approximation is good in the limit of large systems, for which the time scale for decoherence is very short. So MWI-baroque, like CI, is falsifiable, and is in fact false. Like CI, it's false for a mesoscopic measuring device.
So my current view on this is that we should stop talking about the Copenhagen and many-worlds "interpretations" and start talking about the "Copenhagen approximation" and the "splitting approximation" (the latter meaning MWI-baroque approximation).
Allahverdyan, Balian, and Nieuwenhuizen, "A sub-ensemble theory of ideal quantum measurement processes," 2017, https://arxiv.org/abs/1303.7257