Perhaps it has to do with the fact that Asunawa was never elected LCC, and was directly allied with the Senachan, who even then were ruling the fortress of the Light and using enslaved Whitecloaks for forced labor?

Working with the Aes Sedai is nothing new for the Whitecloaks. They whine a lot about it, of course, but they have fought with the Aes Sedai during the Aiel War. They actually helped "protect" their city and center of power.

Whitecloaks fighting among themselves is not viewed with the kind of horror as Aes Sedai doing the same because 10000 Whitecloaks together cannot achieve the kind of destruction and death that ten Aes Sedai can. Like it or not, the contribution of an Aes Sedai is more critical to the Light than that of a Whitecloak.

Which, of course, is why there were two Foresaken working to keep the Aes Sedai split, widening any divisions among them. The WC got no such treatment because they don't matter at all. They are a laughable threat to the Shadow, and left to themselves would have probably harmed the Light more than helped.