Hmmm... I'm not sure where you got that the Holy Roman Empire didn't control Sicily. The Hohenstaufens, by virtue of Henry VI's marriage to Queen Constance, controlled Sicily off and on from 1194 to the 1260s. It never became a stable part of the empire because the Hohenstaufens died out.

The pope very much saw himself as the feudal lord of the King of Sicily, given that the original Norman kings were granted their title by Antipope Anacletus II. (Pope Innocent II was later forced to confirm this grant after being captured in battle by Norman troops) The popes continually tried to assert their authority over as much of Italy as they could. Pope Urban IV actually called a crusade against Sicily and convinced the powerful brother of the King of France, Charles of Anjou, to press it. He ended Imperial control of Sicily.

Sicilians didn't want to be ruled by a French king who they viewed, probably justifiably, as a tyrant. They rebelled against Charles and invited Peter of Aragon to take the crown. The War of the Sicilian Vespers ensued and Sicily was divided. The island itself would be ruled by Aragon and the mainland of southern Italy would become a separate kingdom under the French dynasty until Aragon, now practically combined with Castile into Spain, took it as well in the early 16th century.

Ironically this would result in Sicily passing into the hands of the Holy Roman Empire again, because Charles I (EDIT: not the II) of Spain (and thus King of both island and mainland Sicily among many other titles) was elected emperor to succeed his grandfather in 1519.

I'm not going to go into all the dynastic back and forths that came afterward but I hope it's clear that Sicily was deeply involved in both the empire and papacy.