This may sound like a crazy idea, but...
Is it possible to add a trait file that each family member gets when a new faction leader takes the trone, wich causes a +1 or +2 unrest penalty? The trait would represent the difficulties associated with a new monarch/princeps/whatever being accepted as the new king. The trait would be temporary, of course. Say, 4-6 seasons.
Something like:
Restless people
The new Basileus isn't hailed equally by all his subjects. Some of the more discontent members of the upper strata are probing the water for a chance to rebel. It will be some time before the Basileus' authority is truly consolidated.
Game effects: +2 unrest
EDIT: decrease in cost of bribe would also make sense, I think.
Bookmarks