The POD for this is a failed attempt by Charles Martel to expropriate clerical lands after the battle of Poitiers.
The "expropriation" is essentially a clerical narrative : long story short, it was customary to have nobles trusting lands or goods to monasteries as a benefice, which meant it was for a given period or with likely possibilities to be taken from.
That monasteries tought it was theirs, and unlawful to use it doesn't prevent the large acceptance of such practice ( Late Merovingians practiced this as well), especially giving it was at the benefit of frankish nobility from one hand, and the important alliance between Peppinids/Carolingians and Church.
Furthermore, the frankish clergy is hardly separated from the frankish nobility. We saw Charles put his family and friends in many important charges, in addition of an already established pro-Carolingian faction maybe established since Peppin II if not earlier.
Even admitting Charles doesn't use as much clerical trusts as he did, the redistribution of Neustrian and Merovingian
truste would do just as fine.
The Franks therefore can't suppport adequate cavalry forces to successfully repel subsequent Ummayad invasion and eventuallly fall to the Muslim armies.
The problem with Arabo-Berbers army (rather than Umayyad, as the caliphal authority over Al-Andalus wasn't exactly clear), was they were limited. IOTL, you had only one lasting garrison north of Pyrennes, in Narbonne.
At best, they would have a garrison in Avignon (obtained IOTL during the campaign of 734) admitting they maintain their rule on the city and not just leave it as they did in 726 with Vienne.
But even there, it's likely that Arabo-Berber would loose it even before they loose Septimania : the region was a bit undefendable (straight on the way to Peppinid cores)), and the unstability of early Al-Andalus (Berbers revolts, factional conflicts, Yemenit vs. Syrians, etc.) wouldn't help at all.
It was a piece of cake for Franks to take the region back, and without adressing the whole set of problems there, it would be ITTL even admitting the OP is reachable (which, as I tried to point, is not). Peppinids did well with their own part of
fiscus in the VIIth and early VIIIth century.
Arabo-Berbers forces were simply too few, too much widespread and having front of them powerful and relativly unified forces (at the contrary of Visigothic Spain, not only in the middle of a civil war, but with peripherical regions avoiding entierly royal power).
Finally, you're overestimating the use of a cavalry on Carolingian campaigns : neither in Poitiers, neither at Berre River, Frankish cavalry seems to have played an important role, at the contrary of a strong infantry. Giving that Tours ("Wall of shields"), Avignon (Fortification storming) and possibly Berre River (altough it's less clear, admittedly) didn't seem to involve cavalry use as a main feature...(At the contrary of, say, Toulouse)
Furthermore, contrary to what generally believed trough the sole reading of Byzantine authors, Merovingian cavalry (and mounted infantry) was a thing before Peppinids, as it can be seen trough contemporary elements (conditions of truste, etc.). If Carolingians would have needed a cavalry, it already existed at this point.
I think, eventually, you're not distinguishing correctly the redistribution of benefici by Charles and his heirs to political ends, with a military necessity. Of course, the emerging vassality had a military role, but it served first to maintain Carolingians as redistributors of benefici, which was previously the royal role (hence how Charles Martel managed to be the majordomo of the kingdom, without having a king even as a figurehead).
Merovingian Military Organisation said:
Throughout the last century of Merovingian History, the magnates and their armed supporters became the primary military force in most of Gaul. This is not to say that the number of magnates or even the number of their armed followers increased for no evidence concerning numbers can be found. The magnates and their retainers grew more important militarily during this period because others segments of the military in Merovingian Gaul ceased to exist
At this point, Charles beneficied from the clear dominance not only over Francia (you'd struggle to find possible rivals outside Peppinid lines), but also on peripherical regions such as Aquitaine (Eudon, in spite of participating to the victory of 732, had to renounce being an independent power in Gaul) or Bavaria. Military ressources wouldn't have been a problem.
