"Given that the Covenant is a centuries-old organization that worships tradition and conformity, how do gender and racial identity politics within the Covenant work?"
I find it interesting that when we hear the phrase "hide-bound" or "traditional" in reference to an organization like the Covenant, whose stated mission is killing monsters, we immediately assume that they must be an organization completely dominated by white cisdudes. And I'm including myself in that "we": when I first started working on the organizational structure of what would become the Covenant of St. George, it was extremely old white cisdude-centric.
At one time, that was probably an accurate view of the organizational makeup. While they are no longer particularly religious in nature, or sanctioned by any major church, they did start out with strong church ties, and the church in the Middle Ages was pretty firm on its whole "gotta be a dude with a penis to come to the party." Add in the fact that large portions of the Covenant's leadership settled in Italy, France, and the United Kingdom, and you'd get a very white cisdude party. But here's the thing: the Covenant was good at their job. They still are, as much as they can be; remember that today's cryptids are the descendents of the ones who learned to hide from the raids and the cleansings. One by one, the big flashy monsters were killed off or driven into the shadows, and it got harder to be picky.
There was a time when any village in Europe would have been proud to have a son chosen to work with the Covenant of St. George. By the late 1500s, most of those same villages considered the Covenant a fairy tale, and would never have dreamed of giving away their sons. The Covenant began doing their recruiting from people who had actually encountered cryptids, who could actually see the value in fighting against them. This meant dropping virtually all restrictions against girls serving active roles, because sometimes, it was the girls who survived.
Most of the Covenant's early female recruits provided backup and support for the field teams, having not been trained to fight before they came to the Covenant. At the time, it was not standard within the Covenant for all recruits to receive field training. This changed in the mid-1600s, following the last successful attack on a Covenant stronghold. Half the support staff were killed, as were most of the children currently being housed in that location. After that, it became a prerequisite that all recruits learn to fight, even if they were not constitutionally equipped for field positions.
Resistance to women in the field continued for quite some time, although it had less to do with "women can't handle it," and more to do with "we're not really a powerful political or religious force anymore, and we don't want our male operatives arrested for traveling with women they're neither related to nor married to." Field team marriages became very common, because it was a way for women to get "out of the office" and out doing what they did best. Killing stuff. By the mid-1800s, women had a completely equal voice in Covenant activities.
If this seems like it took a really long time, please compare it to real-world history. The Covenant was surprisingly enlightened, largely out of necessity.
But that only addresses female equality, not race or gender identity. Race was actually addressed somewhat earlier, when the Covenant followed the various explorations, Crusades, and invasions of the rest of the world, either forming or making contact with similar groups around the world. Most local groups were sadly largely absorbed into the Covenant, because the Covenant had the resources and the manpower to make joining forces seem appealing. Those who weren't absorbed are still considered part of the Covenant today, and are not spoken of much outside the regions where their methods dominate. And here's the thing: since the Covenant, and its sister organizations, focused so strongly on "humans first," they didn't bother as much with racial divides. There would be time for those later, when the monsters were all gone. To be fair, if the Covenant had successfully wiped out the monsters in the 1600s, they would probably have turned around and started ethnic cleansing. But they didn't, and they didn't, and they wound up a fully integrated organization by the mid-1800s. They understand racism, and will use it to their advantage when possible, but the Covenant as an institution does not tolerate racial discrimination. Humans gotta stick together.
Sexuality and gender identity are harder, in part because the Covenant relies on "legacy children" for so much of its membership. You're really expected to have kids if you possibly can. While this isn't a law or anything, you'll get a lot of the "Have you tried not being a ________?" routine, especially if you were already slated for inclusion in the breeding program. But at the end of the day, again, they need trained members more than they need to be prejudiced, and have been QUILTBAG inclusive since the early 1900s.
Seriously, there was a time—a long time—where if you were a strong-willed woman who wanted to make your own choices, a gay, lesbian, bisexual, or transsexual teen, or of any race that didn't match the locally dominant racial type, the Covenant was your best route out of that life. If you heard about them, if you heard that they could save you, you took that chance. And the people who took that chance tended to become very loyal, because they were given lives that they could otherwise have only dreamt of.
The modern Covenant is very enlightened, except for the part where anything that isn't human is a monster. There's no room for discussion on that front. And really, that's the problem.