Would result in interesting butterflies. An imperial focus on northern Italy and lack of "lasting" peace in the region means that Barbarossa's son doesn't marry Constance of Sicily, and so Sicily remains under a native dynasty. If the imperial side really does very smashingly well then perhaps the emperor can take tighter control over the north of Italy, though I don't see it lasting very long. The Alps really are a huge barrier to imperial power projection. Frederick II for example spent almost all his time in Italy and the Mediterranean, letting imperial power in Germany erode. It has to be one or the other.
One possibility that I can see is an imperial vicar or scion of the imperial house becoming hereditary king of Italy, nominally under the emperor. Would keep northern Italy under the emperor's control while also giving the Italians their own ruler who could go native in a generation or two, truly making him their own. The emperor would have might control over the pope, and France is bound to attack the empire in any moment of weakness if Philip Augustus still lives and is as brilliant as in OTL. Killing him and keeping the French monarchy weak will make things a lot easier for the empire. On the other hand, France could perhaps (?) reach some kind of settlement with the empire and channel its energies and large population into aiding the Spaniards in the Reconquista or conquering north Africa for Christianity, though this seems likely. A pope that is puppet to the HRE is not a good thing for France.