I don’t think that he simply floundered into the label “a priori” or anything like that. It might have been sort of misleading, but he had a pretty clear definition for it. As he said in The Ultimate Foundation, “[a] characteristic mark of an a priori category is that any different assumption with regard to the topic concerned appears to the human mind as unthinkable and self-contradictory.” For Mises, something a priori is simply something absolutely impossible for us to imagine any other way. I can’t conceive of a universe without causality, because I wouldn’t be there. In other words, it’s “a priori” that there is causality. If you can’t imagine something any other way, it’s “a priori” in that sense.
Now, what’s a priori? I think that it’s simply our volition. As long as I’m trying to study and extend the Misesian project, it’s certain that I’m gonna be acting. Sure, one day I mightn’t be around to act, but as far as I’m still here, it’s a safe assumption to make (that I’m acting or whatever). It would be impossible for me to be wrong about this, because the moment that I’m “wrong” about it, I’m incapable of being wrong about anything anyway. Our volition is a priori simply because we need our volition to think about our volition. We must act to consider our action.
But the question is, “Did Mises understand that the a priori facts were simply pieces of reflective knowledge?” I’m not perfectly sure that I know what you mean by “pieces of reflective knowledge”, but I’m pretty sure. Anyway, as Mises said in Human Action, “The starting point of praxeology is[…] reflection about the essence of action.” Isn’t that the same thing that you’re talking about? The essence of action is what’s a priori about the world (right?), and he’s saying to start out with praxeology we “reflect” on the essence of action. Is there any difference between what he’s saying here, and what you’re trying to say? Or somewhere else did he demonstrate that he didn’t really get it?