Which meant she was there for some time. It stretches credibility that Mesaana entered the Tower, and a week or so later, let slip hr new, top secret position.
But the Black Tower itself did not exist in the beginning of LoC. That is when Rand found Taim, and led him to the farm. The Black Tower as a name was not known to the Tower till aCoS. It is in the prologue of aCoS that Elaida thinks to send an expedition to the BT. She talks of recalling Tsutama, Lirene and Toviene.

The way this has been placed and the way Elaida talks of it, I highly doubt Tsutama entered the Tower before Mesaana's slip.

And in that PoV, she makes it clear that bringing these women back was entirely her idea. If Alviarin was involved in that, wouldn't her immediately following PoV, where she thinks the BT expedition is crazy, indicate that she was frustrated that a woman she helped bring back, Toviene, is now to lead an expedition she disapproves of?

But that still doesn't answer my question. If Mesaana could manage such a big operation like deposing Siuan while not in the Tower, why did she need to enter it thereafter? And if she was there, why did she change who she was later?

This wasn't my argument to prove Mesaana wasn't Tsutama, merely that obscure sisters are not out of the question, as you said in your initial post.
And no amount of division in the Tower can explain factual errors. And no way would torturing Tsutama have revealed all the details. Playing Tsutama, coming in contact with all those Red Sisters and Sitters, is a majorly risky game. Character changes are acceptable, but screwing up facts? Especially with a woman like Pevara, who'd have immediately found out if there was any inconsistency in Tsutama's statement? No...

But she could have hardly known it will come to this when she took on a disguise! She didn't enter the Tower expecting to fail.


Nope. Given that her position might be shortly revealed, it makes every bit of sense for Mesaana to create as much chaos as possible before Egwene gains the stole and can ask everyone to swear on the Oath Rod.

Yeah I love that too.