A very interesting post by Peter J Reilly (Forbes), quoted also by Dean Paul Caron (TaxProf Blog); I'm not an expert on this area of the law, but Dean Caron is. The Reilly post is chock full of details both on the underlying political controversy between pro-transgender and anti-transgender radical feminists and on the tax issues; read the whole thing. Note also that an organization doesn't have to be recognized as a church to be tax-exempt, or for donations to it to be tax-deductible for the donor—many ideological organizations are tax-exempt 501(c)(3)'s; but, as the post points out, being recognized as a church does give one some extra benefits.

I would quibble a bit with one item in the post:

So a cynical bastard might think that this enterprise is a clever way to end run around existing and proposed anti-discrimination laws, because religious liberty trumps those sorts of things which the Supreme Court recently confirmed in Masterpiece Cakeshop v Colorado Civil Rights Commission.

Masterpiece Cakeshop decided only that deliberate targeting of people because of their religiosity violates the Free Exercise Clause, and that the particular facts in the case sufficed to show such deliberate targeting. The case that holds that religious institutions—both churches and other institutions, such as schools—have a First Amendment right to discriminate in choice of leaders (and others, such as teachers, who are closely involved in propagating religious beliefs) is Hosanna-Tabor Evangelical Lutheran Church & School v. EEOC (2012). But beyond that, though Hosanna-Tabor does focus on the Religion Clauses, nonreligious ideological groups are also entitled to exemptions from antidiscrimination laws when their discrimination in choice of leaders and speakers (and likely in choice of members as well) is closely tied to their ideological belief system; that's what Boy Scouts of America v. Dale (2000) held. That, though, is just one detail; again, the rest of the post struck me (and, more importantly, Caron) as very interesting.

UPDATE: Reilly also has a follow-up post.