Can you point out where this is discussed, because I am not seeing anywhere where Joseph and Henri are actively doing what you are suggesting.
I'm not seeing a slander campaign. Before the official announcement, Shade mentions that no one knows what is going on. People were talking about things ranging from a bastard of the King (Joseph and Charles' father) to Joseph's lover, etc etc etc., but, in the examples given, Charles isn't even mentioned. After the royal announcement, it is admitted that Charles fathered twins, and that one of his supporters kidnapped her, getting her and himself killed. None of which is inaccurate as far as Charles is aware.
Granted, behind the scenes, there may be some slandering going on, and we don't know how the conversation between Joseph and Charles a few chapters back ended, but, if so, Henri doesn't seem to mention it; the most direct inference I can see of the actions you are talking about are when Henri writes to Isabella about all of the things the different nobles he encounters are up to. But, re-reading that chapter, no mention is made of that particular noble, Count Balmorre, being allied with Joseph or Charles. In fact, towards the end of the chapter, Henri asks Balmorre about any dirt he has on his rivals, which means that Henri is trying to get as much dirt as possible on as many people as possible, regardless of their faction.