I don't think Elaida let Siuan enough time to gather and send that expedition, but I don't see why being part of that expedition is "reason enough" to avoid Tenobia's advisor? The Amyrlin doesn't get to handpick the sisters on such official missions, just their leader. Several would have been Reds, as would be logical. There's no reason to assume any special loyalty to the Rebels or Siuan among those sisters (had they been sent, anyway).

Why should there be manipulation of Tenobia involved? When Paitar finally told Coladara the leaders meant to go south in complete secret, seven other sisters happened to be there visiting her (the evidence is that they were nowhere where you'd expect sisters hunting Taim to be around the Tower Coup timeline... they were in or near Arafel...) and they all decided to follow the King (and most likely try to discover his real intent and why he wanted to go "meet Rand". Paitar kept that bit secret, by all evidence even his Aes Sedai sister Kiruna has never been told of this prophecy either... it's for the ruler alone.). We know there's no "traitor": the expedition did manage to remain secret until it re appeared in Andor, sothose 13 sisters respected secrecy. Illeisien and her four bumped into Tenobia.

The whole thing appears to be a giant red herring (a string of red herrings in fact). Since TPOD Jordan has wanted us to more than half suspect the Shadow is involved with the Borderland rulers and their going South (in truth, it appears the Shadow concentrate all its efforts on taking control of what the rulers left in the North, not on attempting to infiltrate the inner circles in the expedition). Then RJ worked hard to make us believe the BM were a huge potential problem in Andor re the BT, with loose cannons still in the countryside and Tenobia being a loose cannon herself, and likely to be displeased Bashere was nowhere to be seen. In truth, it turned out they turned toward Arengil, nowhere near Caemlyn just as Elayne had predicted, then made a run for the Hills and Far Madding. They never believed Elayne's tales about Murandy, it seems - or at least never trusted her enough not to send their own scouts ahead for rumours of Rand...

About 2/3 into the original final book (ie: very late in the game), after Rand is able to spot DF by sight, we learn it was all about a secret prophecy that comes as a big splap in the face: yes, the Pattern had a plan B to get rid of the Dragon on the eve of TG, if the Dragon had missed his epiphany and become a huge liability.

Surely, the leaders etc. will play a role in the LB, but the purpose of their expedition south has played out by now. My guess is we aren't likely to hear of the 13 sisters with them again. Brandon merely mentionned them in passing in TOM (we still know the names of only two!). Rand and Cadsuane saw them gathered outside the meeting tent, which post epiphany makes it extremely unlikely there's any DF or BA with the rulers. We have met the Shadow's agents in the top Saldaean circles: they sabotaged what Tenobia had planned for Maradon in her absence.

I'd expect Lan to meet the same problems in Shienar etc.

The most interesting/relevant thing about Tenobia by now is probably the fact she's about to meet Davram again at last, and to be reunited with her runaway cousin Faile and see just how much power that one has gained since... We've been told Tenobia and Faile are rivals. As Faile has long predicted, she's likely to be green with envy (and possibly even rage) when she sees Faile married to a formidable man, powerful in her own right in a web of political alliances in the South and so on.

For the rest, it appears to me "the expedition south" was a multi-purpose device. Jordan wanted to justify Lan's own expedition, and to give a victory to the Shadow on the Blightborder without destroying all leadership in the Borderlands.