Reasoning - using logic to form conclusions, inferences, judgements - is a process defined, taught, and professionally applied by philosophers. So, teach students philosophy and some other 'control subject'. They learn logic, rhetoric, and how to follow complex chains of complex thought.
It seems to be a difficult claim to make, that this experiment would some somehow not show an improvement in critical reasoning - however you operationalize it.
[1] https://plato.stanford.edu/entries/self-knowledge-externalis... [2]https://plato.stanford.edu/entries/operationalism/#OpeUniAna...
The other thing is that you've made a dichotomy out of a quantitative problem: how much better are philosophers at it than anyone else? A little? A lot? Is it a skill that persists after graduation? Better at certain types of problems, maybe worse at others?
In any other context, on any other topic, a philosopher would never dare to make hand-wavey statements about how obvious and straightforward it is that X leads to Y. Only when it comes to their own field of study and its value do philosophers become sloppy and claim the value of their field is self-evident and frankly beyond discussion, or they try to turn it into an irrelevant meta-discussion about the nature of evidence. That's the irony I was pointing out. If you boast about being rigorous, be rigorous even when it doesn't suit your purposes.
You're also conflating rigor with empiricism. It is possible to be rigorous and be a rationalist, a phenomenologist, a theologin, or an empiricist.
By insisting that all philosophy provide empirical evidence, you've misunderstood what philosophy actually does, and assumed the epistemology of empiricism - which has a number of flaws and contradictions (that any philosophical rationalist would be happy to point out).