Hi!
You are on the right track. Even OTL Napi III´s regime was under pressure in the late 1860s. Which was one reason the French were so eager to go to war against the NGF.
Now ATL this pressure simply grows on, since Napoleon cannot find a valve to let it off. The situation inside and outside France is too complicated at that point in history. So when Napoleon dies, before his son can get the throne, the republicans make their move, which is by that point nearly uncontested.