Is metaphysics possible without a faculty of rational intuition?

Of course it is. I am not even talking about "naturalized" metaphysics, the kind of shallow metaphysics that stays close - too close, I would say - to the results of empirical science. I am talking about in-your-face analytic metaphysics of the most revisionary sort, i.e. the work of David Lewis ("other possible worlds are just as real as the actual world"), James Ladyman ("there are no things, only relations without relata") or Peter van Inwagen ("there are no tables or rocks, but there are people and dogs").

Since strong metaphysical claims cannot be established empirically and are also not pragmatic suggestions on how to talk, do we then need to invoke a special faculty of rational intuition - a faculty for which there is no scientific evidence - in order to reach these deep metaphysical facts? No, we do not. For example, the neo-Quinean methodology popular in contemporary analytic metaphysics does not require appealing to any kind of mysterious intuitions and is not even a priori in the traditional sense, as Gideon Rosen makes clear:

"Lewis's case for modal realism is the most striking example of Quinean methods deployed for non-Quinean ends. This is swashbuckling metaphysics that takes us well beyond commonsense and physics; and yet the methodology posits no special faculty of rational intuition or transcendental reflection. For Quine and Lewis both, we start where we are with commonsense and science and refine our views so as to simplify our total theory, seeking economy of laws and basic idioms. ... We are open at every stage to correction as science progresses, so the enterprise is not a priori in any of the traditional senses, and we need not suppose that we have a special faculty for directly apprehending the facts we seek." (Quine and the Revival of Metaphysics, 2014, p. 566-7; my emphasis)

 In fact, on the Quinean conception, metaphysics is not qualitatively different from science:

"metaphysics, properly understood, is just more science. This claim depends on abstract description of science as the search for 'simple' theories that fit the facts, and a more determinate conception of 'simplicity' that privileges economy of basic laws and primitive vocabulary." (ibid., p. 568)

So even Lewis' defence of modal realism, which is surely one of the most audacious and radically revisionary doctrines in the whole of philosophy, could be seen as continuous with the methods of science.

Of course, the Quinean picture of scientific methodology could be false. But even in that case, the methodology of mainstream metaphysics would not be based on rational intuitions. It would just be non-scientific. Since I take science to be dependent on philosophy (but not vice versa), it does not matter, in terms of the field's respectability, whether or not the methods of philosophy are counted as scientific or non-scientific.

There is nothing "woo-woo" about questions of ontological and ideological parsimony, and a normal human brain is perfectly capable of considering them. It was these kinds of theoretical considerations, not an imagined "direct access" to an otherwise invisible realm, that lead Lewis to postulate his worlds:

"If we want the theoretical benefits that talk of possibilia brings, the most straightforward way to gain honest title to them is to accept such talk as the literal truth." 

"Modal realism is fruitful; that gives us good reason to believe that it is true." (both quotes from Lewis, On the Plurality of Worlds, 1986, p. 4)

Lewis' theoretical discoveries have greatly increased our knowledge of the nature of possibility and necessity, the size of reality, the criteria of identity for possibilia, and so on. This is distinctively metaphysical, non-empirical knowledge, and there is nothing mysterious about it.

To conclude: the claims of contemporary metaphysics may be revisionary, but its methods do not involve commitments to revisionary psychology.