I am running a WoT rpg. I have it set around books 3-4 Suin is still the Amrylin. I just cant remember if the White Cloaks had an embassy or outpost in Cairhien yet?
I can't remember either, but the Whitecloaks are present in every country from book 1. As a roleplayer I wouldn't be overly concerned about some of them being in Cairhien regardless of whether they're explicitly mentioned in the books.
Also: you might have better luck with questions of this sort in the WoT discussion section. This forum is usually used for asking questions about the site. <3