Earth staff is the answer.
I don't know if this applies but on Prime Avatars, I tried disconnecting right before winning ...
(The experiment was so I could continue helping people do avatars without wasting time going back to the tuning fork NPC & waiting 24 hours to get another key item.)
Anyways, when I log back in I start out in the Cloister but outside of the BCNM. Then the post Avatar death CS starts. After CS finishes, I come back out but there's no clear time.
I suppose if then if you're only one in the BC, then DC before you win. . .
I'll try this out next week when I get a new permit.
As a full-on attack speed NIN, I am very quick about swapping between WS and TP gear. There's not even enough a split second to macro a new gear set between the time the mammet dies and you exit the BC, so I'm a little skeptical about that explanation.
Edit: DCing is a possible explanation, if when you kill one with a red dot it doesn't zone you out of the BC until you've reconnected.