The discussions about the French kings made me wonder: was there any risk during the French Wars of Religion that France would have spiraled down into political fracturing and loss of royal authority as happened in the Holy Roman Empire?

Quite possibly. Given the frequency of assassination attempts against pretty much every political figure, the dug-in conflict between the Huguenots and the Catholic League and the Politiques in the middle, very little needed to change to set the whole place ablaze.

Henry IV refusing to convert or not surviving any of the twelve attempts on his life would be a main one, given his central role in picking up the pieces.