Your challenge, should you choose to accept it, is to restore the Western Roman Empire with a PoD of after 568 AD when the Lombards invaded and took most of Italy from the Byzantines. To make it easier, you don't have to include all Roman territory formerly belonging to the WRE such as Gaul or the Iberian peninsula. A puppet state to the Byzantines that just rules Italy south of the Po river counts (although you get bonus points if the new WRE includes more formerly Imperial territory).
The Carolingian Empire staying together does not count and neither does the Holy Roman Empire. It preferably has to have a Greek/Byzantine puppet Emperor although any tribal leader vassalized by Constantinople will do as well for me. More bonus points if this WRE, or Italian Empire if you will, outlives its patron assuming butterflies are minimal and the ERE still falls at some point in the Middle Ages.
The Carolingian Empire staying together does not count and neither does the Holy Roman Empire. It preferably has to have a Greek/Byzantine puppet Emperor although any tribal leader vassalized by Constantinople will do as well for me. More bonus points if this WRE, or Italian Empire if you will, outlives its patron assuming butterflies are minimal and the ERE still falls at some point in the Middle Ages.