So I'm in the middle of adding a few things to Italy, and also touching up on the Normans (since they are relevant to the peninsula).
One of the interesting perks of having a vassal pope is that investiture seems works just fine under Free Investiture- you can literally appoint the Pope's successor and this will override the election, at least so far as my initial tests have. This obviates my intended viceroyalty style hack intended to allow an Emperor to meddle in Papal affairs and makes the Pornocracy even more fitting.
So the general idea is that the Pope is now a duchy rank title, but he also holds the Papal States- a separate, de jure kingdom title with Late (rather than early) Administration, meaning the Pope will have an easier time centralizing and of course gets taxes from his feudal vassals (as will you if you usurp the title.) Petitioning for Charlemagne's crown now targets the holder of Patrimonium Sancti Petri, rather than the Pope per se- a subtle, but vital, distinction, as should a local ruler pull a Tusculani and usurp the Papal "Kingdom" they will not only therefore have full power to appoint popes (as the historical Tusculani did) but also emperors, as well as of course having the vassalized Pope with everything that that implies.
As a further effort I have started an Investiture Controversy Event Chain. This will occur sometime in the 11th century onward if the Pope is either Independent or vassal to a Holy Roman Empire, and subsequently all Imperial or Papal vassals, and all Italian rulers, will declare one way or the other (they could, of course, remain neutral, but where's the fun in that?). This gives them traits, with associated opinion boosts/penalties; notably Ghibellines will gain a bonus to feudal vassal opinion and a penalty to city vassal opinion, while Guelphs get piety, Papal opinion and temple vassal opinion. After 1250 if the HRE has Papal Investiture and has lost any foothold in Italy then the Pope is victorious, and the Papal Patrimonium will cease to be a de jure vassal of the Empire, whereas if the Emperor can keep Free Investiture and assert full power over all of Italy (not just the north, but at the least installing a relative on the Sicilian throne) then they will emerge victorious instead, destroying the Papal Patrimonium and ensuring Imperial domination over Central Italy. Note that the Investiture Controversy is a precondition for the Crusades- unless one of Rome, Constantinople, Aachen (or possibly Toulouse or Lyons?) have fallen to the infidels, a subjugated Papacy will not initiate the Crusades.
Finally I intend to create a Norman Conquest event. Assuming the duchy exists and is held by the Normans in the 11th century, then they will examine the Mediterranean for a non-existing de jure kingdom that is not wholly controlled by a single ruler (say a Sicily fully reconquered by the Byzantines), and choose it as the target for a Norman Conquest. Obviously Sicily will be favored, but if there is no railroading or historical conditions are not matched then they might well target some other region entirely. After choosing a potential target Norman Adventurers will periodically arrive with a duchy-rank adventurer cb, while all normans in the kingdom will gain a conquest CB to reform the kingdom. The conquest will end by 1200 if the kingdom has not been formed or there are no more independent normans in the area, and will end once the kingdom itself is created by a Norman.
One of the interesting perks of having a vassal pope is that investiture seems works just fine under Free Investiture- you can literally appoint the Pope's successor and this will override the election, at least so far as my initial tests have. This obviates my intended viceroyalty style hack intended to allow an Emperor to meddle in Papal affairs and makes the Pornocracy even more fitting.
So the general idea is that the Pope is now a duchy rank title, but he also holds the Papal States- a separate, de jure kingdom title with Late (rather than early) Administration, meaning the Pope will have an easier time centralizing and of course gets taxes from his feudal vassals (as will you if you usurp the title.) Petitioning for Charlemagne's crown now targets the holder of Patrimonium Sancti Petri, rather than the Pope per se- a subtle, but vital, distinction, as should a local ruler pull a Tusculani and usurp the Papal "Kingdom" they will not only therefore have full power to appoint popes (as the historical Tusculani did) but also emperors, as well as of course having the vassalized Pope with everything that that implies.
As a further effort I have started an Investiture Controversy Event Chain. This will occur sometime in the 11th century onward if the Pope is either Independent or vassal to a Holy Roman Empire, and subsequently all Imperial or Papal vassals, and all Italian rulers, will declare one way or the other (they could, of course, remain neutral, but where's the fun in that?). This gives them traits, with associated opinion boosts/penalties; notably Ghibellines will gain a bonus to feudal vassal opinion and a penalty to city vassal opinion, while Guelphs get piety, Papal opinion and temple vassal opinion. After 1250 if the HRE has Papal Investiture and has lost any foothold in Italy then the Pope is victorious, and the Papal Patrimonium will cease to be a de jure vassal of the Empire, whereas if the Emperor can keep Free Investiture and assert full power over all of Italy (not just the north, but at the least installing a relative on the Sicilian throne) then they will emerge victorious instead, destroying the Papal Patrimonium and ensuring Imperial domination over Central Italy. Note that the Investiture Controversy is a precondition for the Crusades- unless one of Rome, Constantinople, Aachen (or possibly Toulouse or Lyons?) have fallen to the infidels, a subjugated Papacy will not initiate the Crusades.
Finally I intend to create a Norman Conquest event. Assuming the duchy exists and is held by the Normans in the 11th century, then they will examine the Mediterranean for a non-existing de jure kingdom that is not wholly controlled by a single ruler (say a Sicily fully reconquered by the Byzantines), and choose it as the target for a Norman Conquest. Obviously Sicily will be favored, but if there is no railroading or historical conditions are not matched then they might well target some other region entirely. After choosing a potential target Norman Adventurers will periodically arrive with a duchy-rank adventurer cb, while all normans in the kingdom will gain a conquest CB to reform the kingdom. The conquest will end by 1200 if the kingdom has not been formed or there are no more independent normans in the area, and will end once the kingdom itself is created by a Norman.