There isn't a set of rules around what anarchists can and can't do, but there are some roles that would raise many questions (for example, I would find a hard time believing that someone who was a cop but claimed to be an anarchist was being honest, and I certainly wouldn't trust them or include them in my circles or projects - in fact being a cop would lead me to actively exclude them, whatever their professed beliefs). Another of these would be someone working for a church/religious denomination.
There are certainly religious anarchists - you mentioned Tolstoy, there was also Dorothy Day (who was a founder of the Catholic Worker group), and one could argue that Peter Lamborn Wilson straddles the line between spirituality and religion (I don't know him and am not that much a fan of his work to elaborate, but that is my perception). I've even heard Gandhi referred to as a Kropotkinian-anarchist (I would dispute this on a number of levels, but...) Also, there are groups like the Jesus Radicals, who are Christian Anarchists who, as I understand it, are critical of civilization, seeing it as man exerting himself (genders intended) over God's dominion.
None of these folks, to my knowledge, has sought employment through their religious beliefs, rather, their beliefs around religion have informed/mixed with their beliefs about anarchism. I don't personally understand the fusing of religious beliefs (by which I mean organized religions, ie, those with both set rules, scriptures, an organized priest class, and a conception of higher powers or god) with anarchy, but I know that people do it, and I know people who are pretty right on that have found ways to reconcile that for themselves. Honestly, if they don't get their god on me, I don't care what they do in the privacy of their prayer room. Once someone is involved to the extent of being employed by the church, that creates new questions.
- Is this group evangelical (by which I mean actively seeking converts)? IF it is a religious group that is actively trying to convert people (Jehova's Witnesses, Mormons, some segments of Islam, the crazy people on the street corner with the signs...), I'd treat that with a lot of suspicion and hostility (just like when some commie tries to sell me their party line, or when a lefty anarchists tells me that there is one true lineage of anarchism).
- What is this religions views on issues that might be antithetical to anarchist thought (gender roles, social hierarchies and sexuality are a couple things that jump out as immediate things where there is often conflict)
- In what role are they employed? As you touched on, this might be a big one. I had a friend who was the custodian of a large, liberal church. What that meant was he worked evening after most folks had left, vacuuming, taking out the trash, waxing the floors, refilling paper towel dispensers, and so on. He had a large degree of freedom in that there was no one around to be on his case, and he got paid for it. I don't know if he was religious or not, but it seemed like a pretty good gig. Or if someone worked on a "social justice [yech] initiative" within a church where they had the freedom to organize events and activities that facilitated the intersection of their religious and anarchist beliefs, that might work. Or maybe you get played to be the organ player, that seems like a cool gig, despite working for the church. Once one is in a ministerial role, or that of the hierarchy of the church, I think it becomes harder and harder to justify one's choices.
I don't know if this answer helps much, but it is what I have for now.
