Yeah, while I am satisfied with "sect" due to inertia, it really doesn't describe what a "派" is. Sect is more of a belief based thing, IMO both 门 and 派 should be "Clan", partly because they are very familial structures and partly because there is already precedent for Wudang to be translated as the Wudang Clan...but mostly because Clans within the context of online games are like almost a perfect 1-to-1 match for what the concept of a wuxia 派 is, being an organization of participants that are part of a greater community which is in turn defined by a particular skill, whose purpose is to impart proficiency in that skill for the purpose of competing against rival organizations.
I would think there's a bit of leeway regarding subdivisions within a single Sect/Clan. "Faction" is probably more appropriate for the Huashan case given the political allegory nature of that book. I forget the exact details of the Wuliang Sword, but "Branch" might be more appropriate there.
帮 is a bit more difficult, as while it's like a gang, English doesn't refer to gangs with the same nomenclature. Like you wouldn't refer to them as the Bloods Gang or the Crips Gang, they're just the Bloods and the Crips. I suppose you can meld it with 会 and refer to both as "Association"?
"Escort Agencies" are a little unfortunately named, due to the connotations of "escort", but I can't really think of an elegant title that gets the concept across.