Is saying "measurement" really much more undefined and unscientific than saying "different facets of this superposition of your brain state perceive"?
There are various interpretations that attempt to define "measurement" in various ways, but those are not the Copenhagen Intepretation.
As for whether you can give a scientific account of how data is processed by a brain in a superposition of states, you most certainly can. (Ever heard of "quantum computing"?) It's just complicated.
Edit: I’ve never heard of any treatment of “quantum computing” which doesn’t include the concept of measurement, the Born rule and the projection postulate. Have you?
Edit2: it was maybe not fair to say that the probability of events is not defined in the Everett interpretation because many-world interpretations have addressed this issue since the original paper from Everett in 1957. But as far as I know they have not succeed. Measurement has also been addressed in countless papers and books for almost a century, for what it’s worth.