Were there any significant disputes as to who the rightful heir of the Roman empire was?
Thing is, neither Carolingians or HREmperors claimed to be roman emperors. The closest title you had was the really convoluted title of Charlemagne which was "August Emperor ruling over the Empire of the Romans" (where Romans means rather the city of Rome and the conncetion to the pope), and all his successors including most of Ottonians sticking to the non-descript title of "August Emperor" (Charlemagne himself rather more used his titles of "King of Franks and Lombards" even after 800)I’d say it is already a significant dispute when two people both claim that they are the roman emperor.