This can be a tricky topic, because the disintegration of the Roman Empire is a gradual process underwritten by great changes in theology and politics. The titles and trappings remain in use for some time: Quite apart from Charlemagne, Otto I has himself crowned Holy Roman Emperor in 962, the re-founding of the Holy Roman Emperor that survives until the 19th century. Otto II (r.973-983) in particular passes several decrees from Rome (which survive in the Monumenta Germaniae Historica) styling himself as "Imperator" and attempts to re-establish control over all of Italy.

What exactly counts as "Rome" though is tricky. Augustine of Hippo's The City of God was published in 426 against the backdrop of Roman Imperial decline. Therein, Augustine postulates the Imperial Church as the titular "city of God", which has a duty to maintain the Christian faith and provide unity, stability and leadership even as the secular structures of the Empire fall away. By his logic, Christendom is the ultimate successor state, and to be Christian unites you in the City of God under the leadership of the Pope, regardless of local politics. Such logic holds sway for much of the medieval period at least; in 1066, for example, William of Normandy seeks papal approval before he embarks on his conquest of England.

Romanitas, or the use and adoption of Roman Imperial trappings, stylings and designs, remains a highly important political tool through the early medieval period. Although Bede envisions a gens Anglorum and a unified and politically distinct English state as early as the start of the 8th century, Roman imperialism remains politically important. Alfred of Wessex, for example, fosters very close relations with the pope, and actively appropriates Roman coinage designs for celebratory issues alongside regular English currency. Both he and his children re-occupy and rebuild Roman sites which, in some cases, had lain empty for centuries. This is especially the case in Chester, which was rebuilt by Æthelflæd of Mercia in 907, and rapidly became one of the most prosperous cities in England. In the 920s, Alfred's grandson Æthelstan becomes the first king of a unified England, and following his victory over the Danes, Scots and Hiberno-Norse at Brunanburh, declares himself Rex totius Britanniae, consciously adopting the title from the Latin name of the Imperial province.

Thomas Charles-Edwards' After Rome may provide some useful reading.

Can I ask more narrowly? Focusing on Italy, did Theodoric proclaim/consider himself to be in the Roman Empire?