I didn't suggest Mesaana killed them herself.

I suggested the trio, Mesaana in particular but maybe also Demandred if he is involved with Taim, wanted to get rid of Aran'gar with the rebels. Aran'gar didn't know Mesaana was in the Tower, but Mesaana knew Aran'gar was near Egwene. Mesaana wanted to have freedom to do with the rebels as she intended without Aran'gar meddling, and Demanded may not have liked the idea that Aran'gar would meddle with whatever Mesaana and him intended by bringing an alliance between rebels and Taim.

I think Mesaana's idea to achieve that might have been to bring suspicion on Halima by having the two women linked to her cover story killed with saidin, leaving bodies and evidence behind (unlike what Aran'gar had done when she killed the maids - she made sure to hide their bodies). All she needed next was for the news about a female Forsaken channelling to reach the rebels eventually.

Not a long while later, Narishma came and told the AS about a woman channelling saidin. I don't think Narishma is a DF, I think it's a bit suspicious Rand tought of that warning at all (instead of descriptions of all the female Forsaken etc.) - why would a woman channelling saidin seek to infiltrate the AS in particular, when non-channeller have zero influence at the WT and can only be maids and secretaries and such - and it's Elza who may have told Rand before Narishma/Merise went to the rebels he should warn the AS about this female Forsaken channelling saidar.

As for the murders themselves, obviously, if Aran'gar isn't the one who killed, then Mesaana/Demandred had a man do it, a DF Asha'man. There's full of BA among the rebels and full of male camp followers. Too many for anyone to notice a new one. It would have been extremely easy for a BA to meet the man and show him who the two women to kill were. Anayia was killed away from the main camp - near the privies. Men can't go in the main camp or are very rare there (but Aran'gar don't have that problem). As Asha'man sent to kill Anayia/Kairen would have, though.

As for how Mesaana could have learned about Halima's cover story it's pretty straightforward: Semirhage knew all there is to know about it.

We're told the trio in the WOS used to work together to topple their rivals.

In any case, I'm not 100% convinced about the murders. It could have been an half-baked solution Aran'gar came up with in a panic, as a way to try to make the alliance with Asha'man go away after Delana told her there was nothing she had no argument to bring to make the Hall change its mind, and she may have decided stupidly to target Anayia and Kairen because the two of them were annoying her lately (not enough to kill them maybe, but perhaps Kairen who had just arrived wasn't as satisfied as Anayia with Halima's story of travelling with Cabriana, and was asking Halima questions). The problem is however that if Anayia/Kairen were annoying Halima, it became even more risky to kill them, because sisters could have seen this or even been told by Anayia/Kairen.

In any case, even without the murders there remain the fact Moria came up with the whole thing about allying with Asha'man against the Forsaken - and she is BA - and Nacelle's oh-so-convenient weave to detect saidin (another BA), which made Aran'gar flee (again, Delana left the Hall at that point, she didn't wait for Narishma to reveal his information about Aran'gar that lead Romanda to expose her and Delana. It's the weave that decided Aran'gar to leave, and I think more than the weave she may have figured out she was being set up for the fall by Mesaana by that point. Perhaps, for instance, Delana knew Nacelle or Moria was BA).

There's also the fact Sheriam in particular, but also Delana, were very disturbed by the whole Asha'man thing. Even Delana voted in favour very reluctantly (she was really caught by surprise, she could leave and go ask Aran'gar how she wanted her to vote.. she had to decide on the spot if Aran'gar would want her to vote in favour and remain friendly with the faction supporting this decision to influence them, or if she would have wanted her to oppose it even if the vote passed none the less. Delana didn't foresee that the vote would pass either, she was sick when it did.).

The other BA involved, presumably not involved with Halima either, had no such qualms...