Lothair I, grandson of Charlemagne, dies before his father, Louis the Pious. Then, after the death of Louis the Pious, the Carolingian Empire is divided along the river Rhine between only two of his sons: Charles the Bald in West Francia (which now additionally encompasses OTL Middle Francia) and Louis the German in East Francia. With the now much larger power base in West Francia, Charles the Bald is able to bring under control the rest of the Italian peninsula. Eventually, either him or one of his successors could then march on against the muslim armies of al-Andalus, defeat them, and bring the Iberian peninsula under his control.
That way, the unified Empire as the OP demanded could at least arise; having it last to modern times, however, is another question.