The Augustinian Empire, or what if everything went well for the Capetian during the First Hundred Years' War (and the Anjou stuck with the main branch)
The main point of divergence here is the partial win during the Revolt of 1173–1174, where France could weaken England (in part by giving some English land to allied vassals, such as Kent to Flanders, making it a little powerhouse)
Louis VIII, decades later could invade England during the first Barons war, and proclaim itself king with the help of his father, Philippe Augustus, who gave his name to the "empire"
After the death of his father, he took the throne of France, naming is heir ( the dauphin) as governor in England, tradition that still continues.
With the end of the Plantagenet, Louis VIII could even help is son, Charles the first to secure and protect the Latin Empire from the Nicaean, hence putting them on a leash.
The two brothers (Philippe III being the son of Louis VIII) rule a massive empire going from the Hadrian wall to the mighty city of Constantinople. Philippe III now look at the last rival of France, the Holy Roman Empire, while Charles Ier glance toward the holy lands and the balkans. But the Iberian powerhouse of Castille, Araégon and Portugal are starting to forge alliance against the French behemoth with the HRE, while the Turcs and the Nicean are looming in the east.
Will the Empire survive and become the sole legitimate successor to the Roman Empire, or, like Icarius, will it perish?