@Carp will be better fit answering this, but I'd say that while you had rest of imperial bureaucracy, this was ended by the rise of Carolingian feudality, from which administration/landowning/benefices were almost systematically mixed which was not the case under, say, Merovingians or Lombards (AFAIK in the last case) except in latter periods (and for Merovingians, mostly because Carolingians already dominated these matters)..
To what extent the Lombard bureaucracy represented a continuation of
Roman bureaucracy is something which I don’t really feel qualified to address; it’s rather outside my area of interest. But there is clear continuity of the
Lombard administrative apparatus through the Carolingian and even the Ottonian periods, which is to say that the administrative center of the royal palace of Pavia and its associated class of Pavian
notarii survived not only the Carolingian conquest but the “feudal anarchy” of the late 9th and early 10th centuries. The nadir of the Pavian administration was probably under Hugh, but the Pavian government survived him and was to some extent rejuvenated by the Ottonians who took a keen interest in their southern conquests.
Although under non-Lombard management, the Lombard administrative state arguably survived into the reign of Emperor Henry II, who unlike his predecessors placed relatively little emphasis on Italy and allowed the Pavian government and its administrative class to decay. When he died in 1024, the Pavians attacked and destroyed the royal palace, which had by now become a symbol of foreign control rather than native administration, which is as good as any as a “death date” for the Lombard state. The destruction of the palace was also a important milestone in the dissociation of urbanity and royal power in Italy; Lombard governance had centered on the cities, inherited from Rome, but after the rule of Henry II the agents of royal power were to be found in the countryside, in the castles of the rural nobility, rather than the cities which increasingly would become centers of opposition to royal/imperial power.