If we maintain a large byzantine influence on Central Italy (not only some ports in Southern Italy as OTL) after the 8. and 9. Century, and assume, that Turkish people invade Anatolia in the 11. or 12. century, then a lot could happen in 300 years.
Italy would probably look entirely different than OTL, for example a lesser or no influence of the OTL Holy Roman Empire which would be in TTL probably only the Kingdom of Eastern Francia.
It is then important to find out (As Lampiao said), if the Churches of "England", "France" and "Germany" are in communion with the Roman church.
(Even the formation of these kingdoms could happen entirely different than OTL, therefore the quotation marks)
Then we should find out, how powerful the papacy is, in comparison to OTL. Is he still important for the deeds which confirm the possessions of different monasteries? What about the right of investiture? Does the pope think about himself as the highest ruler of Christianity?
Or is he only a ceremonial head of the church and only important in theological discussions?
How important is the higher and lower aristocracy in the western kingdoms and in the (byzantine) Roman Empire?
What about the diplomatic relations between the Romans (byzantines) and the western kingdoms.
If, for example "Germany" or "France" want to increase their influence over northern Italy, then the Roman (byzantine) Empire is a natural rival, and therefore a support for some kind of crusade from these kings is unlikely. But maybe the Emperor don't want foreign kings in his territory, but the support of the European Nobility.
If there is some common christian identity in Europe, and the possibility to conquer land in Anatolia (as vassals of the Emperor) then support from the Higher and Lower Aristocrarcy/Nobility is possible.
The support of the Papacy for the Empire is of course really important for the Emperor in such a critical situation. But depending on the circumstances the call for some kind of "crusade" could either only bring some few fanatical peasants or in the best case a "real" "crusade" with substantial support towards the empire.
But if the pope want to gain more independence from the Emperor, he could also decline any support to the Emperor, arguing that the defeats by the Turks are a sign of sin of the Emperor. This scenario would lead instead towards some independent papacy with control over large parts of Italy.