I don't think Mamizou even lives at the temple. Ichirin doesn't recognize her in HM, and even goes so far as the say that no one at the temples uses techniques like Mamizou's. It's unclear if Byakuren recognizes her or not. And whenever Mamizu shows up in later works she's always off doing her own thing.
The simplest explanation is just that she never accepted Nue's invitation.
Prologue of TD Extra and Youmu's Extra story shows that Mamizou took an interest in the temple at least, and even rested there a few times.
About that part with Ichirin, she honestly thought FakeMami was Byakuren untill it is made clear from the technique she used(but she didn't know she was Mamizou, till she undo her transformation). However, this doesn't mean that Ichirin doesn't know Mamizou, since right after that Ichirin got mad at Mamizou for faking herself as Byakuren. So, she knew about Mamizou and her abilities, but haven't exactly experienced it first hand. She also knows that Mami is a friend of Nue from her VS win quotes. However, this probably means the she's not close to her.
As for Byakuren, she recognized her real identity right after they fight, even before Mamizou undo her transformation. This pretty much means she knows her quite well, although never fought each other before since Byakuren compliments Mamizou's transformation abilities. Seeing how they interact with each other, I get the feeling of them being "allies". Not seeing each other as enemies, but not so close to say they're in the same group.
So Mamizou doesn't exactly practices Buddhism, nor represents them, but I imagine she stays around there from time to time.
Nue however, seems to be taking the disciplines. Just a guess since she's really getting along well with everyone in that second time the Myouren Temple crew got a spotlight in Forbidden Scrollery.