Yeah, but, I think it's a "foundation".
Wouldn't matter. The C Street House is also being run by a "foundation." I'm not sure whether they are going by the name of "The Fellowship" or "The Fellowship Foundation" these days, or whatever moniker they've come up with recently. But they ran into problems stemming from all of 4-8 senators renting rooms from them while they were staying in Washington, DC.
What matters is what is declared on the books for tax purposes. If the property is declared tax-exempt due to its alleged use for religious purposes, owner / management of the property can not also at the same time be collecting rental income from
residential use of the property.
There are probably small local differences in how jurisdictions interpret this, but I imagine the general principle is pretty universal.