No, neither one knows.
(Military men and administrative men follow separate chains of command.)
Though, I may be misremembering, but I don't think Peering gives you any one specific piece of information you seek. I think the spell you're looking for is the lower level Posing the Silent Question; the motivations and personality quirks of every schlub in the camp aren't terribly relevant.
((Well, I said that as kind of an RP piece, not a game piece. Mention his name, to get the thought come to the surface.
Peering into the Mortal Mind allows Sophronia to probe and understand the contents of the mind of the target, including his immediate and long term motives, personal strength and weaknesses and other pertinent information. The Base for the spell is 25, and the only guideline with that base is:"learn all you wish from a person's mind." So did Braith know where the prince was at, or did he suspect that someone else might know where the prince was at?))
Braith did not know where the Prince was, but it was quite clear that he knew the Prince was with a great oaf of a knight who likes to wear red headgear.
Meanwhile, back at the Mercer House...
The party is going in full swing. Attravere was invited to join them when he came in after the harrowing Council Meeting to sneak a bacon sandwich from the kitchen. He is being something of a wallflower, despite the conversation being in Latin. Aequi may be present as well; if chico ever shows back up, lurking about during the party might well be why he missed the council meeting.
Guillaume and the other Redcaps are wining and dining with Prince Thomas, Captain Rhys, and Brother Pryderi. (Only Great William is missing; even Parsifal has taken a seat and is enjoying a mug of Sansa's Very Special Ale.)
Meanwhile, back at the Mercer House...
The party is going in full swing. Attravere was invited to join them when he came in to sneak a bacon sandwich from the kitchen, and is being something of a wallflower, despite the conversation being in Latin. Aequi may be present as well; if chico ever shows back up, lurking about during the party might well be why he missed the council meeting.
Guillaume and the other Redcaps are wining and dining with Prince Thomas, Captain Rhys, and Brother Pryderi. (Only Great William is missing; even Parsifal has taken a seat and is enjoying a mug of Sansa's Very Special Ale.)
Alright, point taken. Headgear is so generic as to mean something completely different. I was thinking of a red helmut.
Sheesh, when you swing that clue-by-four, you don't hold back.
Oh, and she makes her way to the Mercere House.
I thought I'd swung it pretty hard the first time 
Well, it is character familiarity. If you had said a red cap, and not head gear, I would've put it together much faster. Head gear makes me think of a helmut, like hockey, football, boxing, something athletic or martial.
Sophronia finds a contingent of armed men hanging around outside the Mercer House, raucously drinking ale and munching honeycakes.
Inside she finds a more intimate but equally raucous party, as no one parties harder than Redcaps and nobility.
It was "red hat" the first time 
"Cap" makes me think of Steve Rogers first, and Babe Ruth in a stupid looking hat second.
Language is a wonderful thing. Those are closer to what I envision for redcap. Red hat makes me think of a more modern type of hat, like a stetson bowler, top hat, something like that.
Or a fedora 
I picture them as the kind of hats they wear in movies set in the 20's. Like in Newsies. I don't even know what you'd call that.
Well, umm, considering they were newsboys, I think the term for those would be newsboy caps. 
So, back on topic, Sophia, Attravere, and possibly Aequi are hanging out in the Mercer House with the Redcaps, the Prince of Gwynedd, and a couple of the Prince's men.
The Prince speaks Latin flawlessly, in the way only Oxford-educated men do (his perfect lack-of-accent is as distinctive as an actual accent). Brother Pryderi speaks Church Latin quite well, and the Captain stumbles along as best he can.
The current topic of discussion is a debate on whether or not Galfridus' writings were historically accurate. Thomas is arguing vehemently that they are primarily fiction, while Guillaume argues fervently that they are largely biographical, with fiction limited only to areas where myth has distorted the truth. There is some divergence as they argue objective reality and whether "fact" can exist independently of belief and perception.
They're called berets. Like the Prince song.
((I was expecting a whole shack shimmy, not a philisophical discussion.))
Alright, Sophronia's going to stick out like a country bumpkin with that Latin of 4. She'll hang near the Captain. Maybe get a little cap'n in her.
The Captain appreciates the female attention, but he is quite self-conscious about his poor Latin, and seems quite embarrassed as he speaks.
Is Sophronia drinking what the rest are drinking?
Assuming no further intervention, the archers are conscripted into Prince Thomas' army, and they march forth the next morning to war with Prince Llewelyn.