Well that's not completely true. As I explained in my Visigothic timeline, the first 'suggestion' for help came at the beginning of the 7th century when the Popes were worried about the effect of the harsh Byzantine - Sassanid war. And I think that since then the Popes stopped to rely on Byzantine protection, which is more than 150 years before the actual Carolingian assistance.
However, Visigoths were unable to assist the Papacy in Italy and the Franks were involved in a Neustrian vs Austrasian internal war, so the first demands of the Popes were neglected, and this lasted for 150 years more.
If Charlemagne would just have prioritized his internal issues over the Roman pleas for assistance, the intervention in Italy would have just not happened and maybe the Lombards would have taken Rome, but as long as they were mostly Catholic by the time I can't see the tragedy here.
The Popes were pretty much appointed by/dominated by Constantinople from 534 to 752 (the era of Greek Popes)... The big split began when the Romans confiscated papal patrimonies in southern Italy in the 730s. This would lead to the fall of the Exarchate of Ravenna (the Imperial stronghold), and lead to the Lombard occupation of that territory in 751. This was due to disagreements between the Papacy and Rome, and would eventually lead to the invitation to Charles some 50 years later. So Rome was inextricably connected to the Papacy until the fall of the Exarchate...
Do you have a source for your Visigothic claim? Thats the first I have heard of it. It obviously didnt last or pan out, as the popes asked for Imperial approval till 741.
And with regards to Charles focusing on internal issues, his internal issues would be taking the rest of the Empire from Carloman. Without the resources held in his brother's Kingdom, Charles will be hard pressed to do much... But that would lead to more Romanization, something you are trying to avoid. The brothers got really close to war before Carloman died, which could have gone either way (as we have little idea what Carloman was capable of).