I mentioned earlier that the Lombard kings styled themselves as "kings of the Lombard and Italians", and probably did not have a full grasp of the concept of territorial state.
Even if they had, it wouldn't have mattered. Up to the XIII at best, kingship and lordship is really about popular and/or individual relation to their ruler (by exemple, with the dynastic change in East Francia, the title passed from "King of the Eastern Franks" to "King of the Teutons").
I'm not saying that territorial rulership was unknown, the counties were more or less such, but in regard of kingship it's simply irrelevant.
However both of them managed to keep this submission quite nominal, and their policies make it clear that they saw it as a continuation of the strategy of the dukes of Benevento (the largest Lombard duchy and the farthest away from the royal capital in Pavia) toward the Lombard kings.
Well, certainly. On the other hand, Carolingians make it clear they saw their rule in Italy as a continuation of the Lombard kings. I assume Benevento's policies were quite normal on this regard.
It is interesting that neither Charles nor his son Pepin of Italy ever managed to crush the duchy completely: certainly the distance helped but also the fact that Campania had been promised to the pope as part of the Patrimonium Petri (and it was mostly Adrian I who insistently asked Charles to move against Arechis
I can't remember a lordship that Carolingian crushed completly : they favoured keeping the structures already existing but putting pro-Franks and Franks nobility in charge locally and Frankish rulers.
Now, they never managed to do that for Benevent. But should we go as far saying Benevento wasn't under Carolingian overlordship even if they tried to dispute it? I don't think so.
For the Donation of Pepin, I'm far more dubious than you : it was notably promised to papacy the control of
Tuscany and Emilia but it was never enforced. I don't doubt roman papacy tried to take advantages of frankish expeditions to expand (or more exactly, trying to expand, itself) but the reasons could be more secular : Arechis participated to a plot with Lombards nobles under Frankish rule, and the having a southern principality openly rejecting frankish suzerainty was a bad exemple for Lombards nobles.
Finally, a struggle of influence against Byzzies is another possibility, but I would more tend to see Grimoald doing see-saw with both empires, using his position as a buffer principality.
All of this is certainly interesting (and I was always fascinated by the history of the Lombards in Italy) but the point I was trying to make is that the Carolingians never went beyond Benevento (actually they never even reached Benevento in arms)
As I said above, they did. I can list three armed interventions in southern Italy.
-787 : up to Salerno in order to sumbit Arechis
-788 : allied with Grimoald and Hildebrand of Spoleto to fight Byzantines.
-792 : to fight Grimoald that tried to overthrow Frankish suzerainty.
However, with Louis I, the situation is far better for Beneventi as Carolingians are too busy fighting each other or dealing with Vikings to make the former forced to do only really nominal gestures of submission.
and recognised a de-facto overlordship of the ERE in Southern Italy
Erm. No. Unless fighting Byzantines and their Lombard puppet Adalgis is a sign of recognizing ERE overlordship, they didn't.
To my knowledge they were under Byzantine influence in the period between Islamic invasions on southern Italy (late IX century) and the definitive takeover of Italy by Ottonians (late XI), but not during the Carolingian era proper at the exception of really late part.
This played well in the hands of the popes a couple of centuries later when the Byzantine presence was even more reduced (and Sicily had been lost to Arabs) to feel entitled to grant Roger the Guiscard the crown of Sicily, then later on to appoint an Anjou as king of Naples (doubly justified I would say since Campania was included in the Patrimonium Petri) and even later to create a kingdom of Sardinia and to offer the crown to Aragon in exchange for an end to the hostilities with Angevins in Southern Italy.
If Charles had not agreed to the idea of granting almost half of Italy to the Patrimonium Petri (a nominal grant, obviously, and the popes never managed to get their hands on most of it) the history of Italy would have been quite different (and the history of Europe too).
I really doubt it played a role. Not only popes never managed to get Emilia and Toscany but even in the regions as Ravenna or Pentapolis, they barely managed to make their nominal authority respected.
The pope was only considered then as the administrator of Rome and the nominal overseer of the former exerchate.
If you search what created not only the pontifical temporal independence but as well control of Central Italy, the Ottonian Pact is probably far more fitting as it clearly specify these.
OTOH Charles' father got a very useful investiture from the pope when he finally took formally the throne, so maybe Charles was trying to pay a debt.
That's, if you allow me, is really far from carolingian relation to papacy. He needed it as a tool and nothing more : the frankish king was really clear on this, the pope had to obey the leader of Christiendom, him. The role of the roman pontiff was to be busy supporting him with prayers and christian unity.
You only had a more balanced relation between emperors and papacy with Louis I, admittedlty raised as a monk, that had both a better respect for the pope and a need of legitimacy against his sons.