3. Question: She could swear that she’s not a Darkfriend on the Oath Rod, right?

3. Answer: As long as she believed it to be true. Every remaining Aes Sedai in the Tower has retaken the three oaths. You should be thinking about ways to defeat the Oath Rod. There is a way to do it.

To me this is Brandon nudging our shoulder that the answer lies in Mesaana being able to defeat the oaths, and NOT a mind trick about her not really being a lowly Darkfriend.

I have read through theories from "she inverted gloves of air around her hands so her skin wasn't touching the rod" to "She could technically say she is Chosen and therefore not a DF".

My personal theory is this (influenced strongly but not entirely by Luckers over at Dragonmount):

Mesaana was masquerading as a "random" AS that she had already interrogated and killed. We know she was in contact with Sheriam in the rebel camp, so presumably it could have been Mesaana herself who warned the BA sisters in the tower. Saerin and the other Sitters had to take a census of Sisters and initiates the day after the battle, when Egwene was cleansing her Aes Sedai. Mesaana could have used the confusion of nobody knowing who was still within the tower to change disguises. All she has to do is select a new AS, imprison her in her own rooms tied up with inverted flows of air so nobody will sense her or disturb her. Fail to show up to the census as her old disguise (presumed dead or captured). Then, when Egwene is raised in the Tower, knowing that the second round of cleansing is about to begin, compel your prisoner and send her to reswear the oaths with orders to return to you when it's over.

I personally think this is one of the most plausible theories (quilted together from other ideas) especially since it doesn't involve any tricks of channeling or knowing how to block the binder. It's a simple "switcheroo".