How early is early - Anything before 850? 800? 750?
To be contemporaneous with the march from Gibraltar to Tours, it would need to be done before 720?
I think before 800 would be ideal, but it is hard to see how that would work given the conquest of Sicily didn't start until 827. If you can have a significant Muslim presence in Southern Italy by 750, you would see a totally different papacy, a completely different middle ages. It would be a fun TL to write - what happens to the Rome-centric view popularised by the island missionaries when Rome is threatened (or engulfed) by the heathen menace? Where does the Carolignian dynasty turn for legitimacy? What happens to the claims of Milan when Rome loses almost all its revenues?
But I fear realistically, your best option is an earlier and more complete loss of Byzantine holdings followed by a long war with the Lombard duchies in the mid-800s. The Carolingian Empire is in political decline, the status of the papacy in question again, and there is nobody who can realistically stop the invaders getting a foothold. By the time the Ottonians come to Italy (if they do ITTL), there could be an established Muslim state. With resources equivalent to those of Norman Sicily, it could certainly hold out while.
What if the early invasion captures and holds Rome for a generation or more? Would that lead to a fervent Crusade or Reconquista movement no matter how early it occurs?
It would have to happen at a time when Rome does not enjoy the reputation of being the centre of Christendom. The medieval papacy engineered that idea very effectively, but without the support of the Carolingians, it would have been of dubious legitimacy. If the Muslim state holds Rome either during the reign of Charlemagne or during the early Ottonian years, these rulers will have to look elsewhere for a foundation to their crowns. As a result, the key role of the bishops of Rome may become far more understated (a kind of theoretical final court of appeal rather than the active head of a universal church). In that case, the capture of Rome would be of not much greater significance than that of Naples, Palermo or Beneventum. You'd see a reconquista, but not a huge crusading movement.
If it happens when the papacy already has its medieval structure (under Charlemagne, who is invested hugely into supporting the popes, or after the Ottonians), you would see a massive military effort to retake Rome. Probably successfully, given the mismatch in resources.