Do you think that the best way achieve solutions to meta-philosophy is to actually do philosophy?

I don't know, but I think it's at least plausible that the answer is yes. This is one of those situations where we should probably take multiple approaches simultaneously.

The issue I see is that to giving a satisfactory answer to a question like "what is the nature of philosophy, and how do we use it?" likely involves a whole hell of a lot of neuroscience, psychology, linguistics, etc. research [...]

Maybe, but before von Neumann and Morganstern invented expected utility maximization, it might have seemed like we'd need a whole lot of neuroscience, psychology, linguistics, etc., to figure out decision theory, but that would have turned out to be wrong.