Does King's Landing know about Stannis?

[Hammermen's Revolt]

I just realised that I didn't recall any incident of the KL folks (tyrell or Lannister) say anything about Stannis rallying the north. Do they even know he's there? As Bolton knows, it would seem obvious he would've sent a raven, wouldn't it?

I do remember the surprise about dragonstone being so empty, but that's all I recall. If they know, does anyone know what their tactical plans are regarding Stannis?

[Beorn Snow]

They know he went north but do not consider him really a threat I think.

[Consigliere]

Ser Kevan knows that Stannis is in the North. I doubt KL knows the exact details of whats going on it the North though.

Ser Kevan - “Stannis in the north, ironmen in the west, sellswords in the south.

[Fightbringer]

KL does know, aside from the evidence from Kevan, Jaime thinks something like 'Stannis is welcome to the wall' in one of his aFfC chapters. They must at least be aware that he is pursuing plans to fight against the boltons, as Cersei and the small council dealt with Wyman Manderly when he had Davos, so they must have known what that meant, if he didnt tell them. I'm not sure whether they know the current state of things, they probably dont know about the blizzard, but would think they know about Deepwood Motte by end of aDwD.

I think the main thing is they dont consider him a threat, as he has significantly less than 10,000 men, perhaps even 5,000, thats enough to be a threat in the North as things stand, but the IT has ten times that number in the South, so he's no threat as long as he doesnt gain the northern forces to his side, and also gains the backing of the Iron Bank (which i'm unsure if KL foresees, Kevan might have, but Cersei didnt and Mace won't) and even then they have a good chance of winning if they can keep there alliance with the tyrells going. So a few things have to happen until he is a real threat to the IT, and the Ironborn in the West and those Sellswords who are landing in the Stormlands are of greater priority since there closer to home, and both currently have more men than stannis does.

[Thendel]

His name was thrown among the commoners during Cersei's WoS, so he definitely hasn't been forgotten entirely. The fact that he still holds Storm's End and until recently Dragonstone probably hasn't gone unnoticed, given that Tyrell soldiers were dispatched with much fanfare to take these two locations.

That said, I doubt anyone really knows about Stannis' movements during Dance: neither Stannis nor Roose really has any interest in word sending word south of the Neck: Stannis needs to get the North behind him before going south, and Roose is most likely banking on the current leadership crisis in KL to keep focus away from him until he is ready to proclaim himself KitN.

[MotherAnduin]

Even one of the dumbest character in the book, Obara knows that Stannis is in the North. The people in King's Landing are not that incompetent.

[Stannis Eats No Peaches]

Even one of the dumbest character in the book, Obara knows that Stannis is in the North. The people in King's Landing are not that incompetent.

Famous last words...

[John of Mowsley]

They know he's in the North, but they don't know what he's doing in any detail.