The Italian campaigns were finished in 556. The three chapters schism between the emperor and Pope came to a head. Some popes were dragged to Constantinople. This shows that the Emperor was exerting influence in Rome. The Lombards invaded Italy in 568. In 590, the Vatican erected the sign of St. Peter, symbolizing its independence and the creation of the Papal States.
The Exarchate of Ravenna included a small strip of land, through Perugia to Rome. This is how Rome survived the Lombard invasions, but was isolated enough to be independent. It appears to be a fine line; the empire could maintain the "bridge", but couldn't tell the popes what to do. I'm wondering if they actually tried to occupy Rome, that they couldn't have handled both the Roman and Lombardian response, and would have lost other Italian regions. For example, when Justinian II tried to arrest the Pope, the people in Ravenna and inland rebelled, before they even got to Rome. (Although, I like T.E.D's story of hiding under the Pope's bed more.) On the other hand, if Rome wanted assistance against Lombardia, then constantinople could assist Rome via Perugia, but then Constantinople would have wanted something in return?