Charlemagne would be butterflied away by a complete Lombard conquest of Italy, and different interests in the Byzantine empire might even stop an Arab invasion.
It's difficult to say what might happen, it's going to depend on how long Alboin lives, and if the succession is disputed or not.
The Lombards would assimilate even more quickly, and most likely convert to Catholicism. If the royal power is strong, as it might be after a long reign of Alboin and an undisputed succession, the kingdom of Italy might endure.
Probably a pawn of the Lombard king, during their best time maybe something like the Patriarch of Constantinople during the XI century. To let the roman Bishop acquire even more power, especially outside of Italy, would allow the establishment of dangerous international ties that could pose a threat to Pavia. Think of it as the Roman Senate during the VI century and its tense relationship with the Goths (due to dubious loyalty), except now with the entirety of Christian Europe.
So basically, yes. If the Kingdom of Italy is up, the Pope is down
their legacy however might lead to Italy following a similar historical path to France, that of a state whose history as a coherent unit headed by one king dates back to the Early Middle Ages, despite the endless disputes between the crown and the feudal/local level.
I see two doors ahead for Italy and Europe -
Through door #1 later foreign wars and foreign invasions and gross fragmentation weaken and break the cohesion of Lombard Italy as the cohesion of Visigoth Italy was broken, and there is a chance for something somewhat like OTL's multiple powers, city-states, and the medieval papacy to arise.
Through door #2 the Lombard Kingdom of Italy endures, Italy follows the historical path of France, the Bishop of Rome is the creature of the King of Italy in Pavia, Ravenna, Rome, Firenze, or wherever. The absence of any equivalent to OTL's Papacy if a HUGE divergence. It's Caesaro-papism and Church subordinate to Emperors and Kings everywhere, everybody is Orthodox, east and west. There's probably less of a bar to using vernaculars for church affairs in later centuries. Missionizing is probably far less intense, except where it directly coincides with existing Christian states diplomatic interests. You'll have individual missionaries, and monastic missionaries like the Irish-based ones and others, but not the Rome-Papacy centered global missionary effort. Paganism can survive longer in central, northern, and Eastern Europe.