Suppose Latin remains the language of religion, learning, and diplomacy. And suppose Rome itself remains home to some sort of spiritual leader, a "father" if you will, a successor to the high priests of antiquity. Heck, he might even keep the ancient, pagan title
Pontifex Maximus. He can act as an arbiter between the various breakaway kingdoms. Maybe that "Pontifex" is even the head of a huge network of Latin-speaking religious officials, also called "fathers," who sanction the legitimacy of the secular rulers, act as the keepers of high culture and ancient texts, maintain large estates and priveleges, and in general help maintain a unified, Roman culture throughout the Western world. Depending on how that religious/cultural network plays their cards, they could have a very long-lasting and influential system!