Shane, I regrettably don’t have time to look this up in any detail, but as an educator I suspect that there may well be some experimental work already reported about causation v correlation in the education literature, particularly in the fields of engineering education and nursing education.
Universities can be good places for piloting such research. For instance: a pre-test in a course that teaches critical thinking methods, and a post-test that appears as a pre-test in some subsequent course. You don’t want to have a post-test within the critical thinking course because the students will be primed to use that knowledge; what you want instead is to see if they can use it without being primed.
Not exactly what you seem to want to do, but it might help point you in a good direction and possibly help you find collaborators.
Anecdotally, I’ve found the biggest problem is students are getting to University (engineering, in my case) without ever having been taught any critical thinking at all. There’s 2 problems there: the primary/secondary schools are being negligent, and the universities are accommodating their negligence. This is to say that the problem is significant, widespread, and, thanks to the politicization of education, unnecessarily complex.