The Great Caliphate
Their conditions were met, in Otl, at least for one or two centuries after the last of the Umayyads took refuge there, and form an independent powerful and prestigious beyond its borders Emirate.
But if what you want is that this situation is permanent and even higher than in Otl, avoid the Christian reconquest of the Franks (took Barcelona in their invasion), or in the case of Iberian, reducing it to incursions of mountaineers, little more than bandits of the Cantabrian Mountains.
Avoid becoming isolated political and culturally, behind the Pyrenees; for that would be needed during the siege of Toulouse (Aquitaine), 9 June 721, Duke Odo made a surprise attack from the al-Andalus, Wali (governor), Emir As-Samh ibn Malik al-Jawlani and defeated him in the battle of Toulouse,not surprised or managed to take the city or negotiate his surrender before the arrival of the Christian Forces.
In turn this defeat led in Otl, to a subsequent incursion, was famous when It turned triumphant Charles Martel and allowed him prestige and legitimize his usurpation of the throne Frank founding his Dynasty that ended consolidate the Franks State.
If the result of either Battles, would have been a victory, the Islamic dominion, over at least part of Aquitaine, we have given it a buffer zone with the rest of Gaul, avoiding the formation of a powerful a Christian state north and mastery of the Pyrenean valleys, the Bay of Biscay ports.
This domain probably would cause a faster Islamization of Basque Country, Septimania and at least the coastal regions of Cantabria Iberica.
The latter would prevent the formation of the Kingdom of Asturias and therefore the beginning of the 'Reconquest' Christian.
The Al Andaluz was already a very powerful and influential military, political and cultural power in medieval Europe, Otl, for many centuries.
In my scenario I suggested, without internal frontiers, with a trans-border from Arles to Bourdeaux, with a growing population and a constant process of Islamization, would have no trouble maintaining and exceeding the levels reached in Otl.
An Al Andaluz increasingly powerful without the threat of a United Frank (without the Héristal or not so powerful), weak, disunited, in which each time a king or nobleman important dies, its territory is often divided among his heirs, they see their power and influence increasingly diminished.
In conclusion: a kingdom of the Franks in the state I have described above and more Germanized.
A Lombard Kingdom in Otl was more disjointed and weaker than its northern neighbor, which would be isolated and its center of power, the Padane plain threatened by a vassal of the Emirate: Provence.
It would surely conquered in the next inevitable wave of expansion Islamic , supported by own resources and by native peoples for newly Islamized (Basques, Iberians, Gallo-Romans), led by Arabs and Berbers with their logistical lines stronger and safer.
The fate of Italy with the Mediterranean coast of Gaul and the Alpine steps to the Caliphate, would depend on the Byzantines, who had their own more serious problems in the Balkans and Persia.
The Papacy that before the Islamic Threat, would be more politically docile with the Lombard Realm and the Byzantine Empire, politics and theologically, become a Patriarch elected with approval of the Emperor.
Rome probably would be a Byzantine exarchate, if they have enough troops. If they can not, it would become the capital of the Kingdom Lombard, at least while they can defend it, from the growing threat by Sea and Land of the Caliphate of Al Andaluz.