so, there is no neutral path between the family members, you either go with cath and mom/elin don't trust you or you kill cathIf you don't kill Cath at the Tavern the coup happens no matter what. If you say no to Cath initially she'll lock you in jail and then ask you again.
From what I understand there is the Green king path, which is what happens when you either side with Cath either when she initially offers or when she asks you later in prison. There is the Red king path which is what happens if you side with side with Cath and then have Isis poison her, then there is the White king path where you reject her deal all together and break out of jail and eventually get supports to stop her coup. There is no way to stop the coup from happening all together. I only really been playing the Red path and the Green path and haven't noticed any difference in Chapter 0.8 other than Cath is gone from all the scenes on the route where Isis kills her.