By using 'republic' for the Roman Empire in general, authors of the period simply perpetuated the classical and post-classical name for the Roman institutions and state, with 'imperium' being more easily a way to consider the king of power emperors held on and in the state. Although it would get mixed along the way, the definition of the Eastern Roman Empire as a 'politeia', that is a state, remained well after.
More than just a set of institutions and political exercice, a republic also carried an ideal of good administration and governance, under the influence of Cicero's works for instance, a political and moral tradition carried by the Romans, or ones they ideally should.

On the other hand, kings in the former western provinces were held ruling 'nations', that is peoples : a Frankish king ruled Franks in particular but also happened to manage things in former provinces ideally (at least for Romans) 'in unity' with Romans and the republic.
This was largely coming from the political fiction of the Vth century in which Barbarian kings were seen (or even saw themselves, as with Gondemar's letter to Anatasius where the king call himself a mere "soldier" of the emperor) as subservient to the emperor or at least collaborating with the Roman state (as with Cassiodorus' eulogy of Eutharic as restorator of the republic in Italy) : Constantinople granting not the royal title that depended from the Barbarian peoples themselves, but state functions and honours (consuls, patrices, magister militium, etc.).

Of course, this didn't stand much against realities of power, and by considering themselves as 'princeps' in their realm, these kings were effectively the heads of their own public service and states.
But the distinction between 'nation' and 'republic' was kept both by the Empire (cf. letter from Maurice to Childebert), whereas it was accepted the kings effectively beneficed from their proper 'imperium', and by western writers themselves set in a whole range of classical and post-classical references : the Roman state was the republic, even an 'holy republic' as it's termed in papal letters.

The use of 'republic' in 730's to 750's century Central Italy was divergent from this, as the pope Gregory III used the same term of 'holy republic' for the territory he effectively temporally ruled in Latium, headed by St. Peter and distinct from the 'holy republic' led by the emperors. This was equalling them in prestige and authority, being particularly relevant in a period marked by religious divergences between Rome and Constantinople but also renewed Lombard takeovers in the region. Eventually it led to a more systematic use in pontifical sources of these papal estates as a republic 'of the Romans' in Italy, that is as a ordained state identical with the Roman Church, with implication of some domination (either regional or religious).

This peculiar usage didn't change much things in Frankish chronicles, and still under Charlemagne the eastern Empire was called a 'republic', for instance in the Royal Annals although they have no qualm granting both Charlemagne and the eastern emperors as exerting an imperium, and the former being emperor and August.

A different use of 'republic' happens in the IXth century, after the reign of Charlemagne with his son's : as Carolingian Francia underwent a severe political, military and succession crisis, references to imperial apparatus and prestige became more present in texts, the empire never existing as much it did as it broke-away : the 'Republic of the Franks' (Walafrid's prologue to Vita Karoli Magni) and mentions of 'the republic' for the Carolingian ensemble (Nithard, Metz Annals, etc.), an use that did not outlive its collapse.

When Gregory of Tours used 'republic' for the Roman Empire, that for him did not cease to exist but just relocated and reconfigured in the East, it was not just about making a geographic or ethnographic point with the true republic being the Roman Empire, but also a political one addressing an idealized perception of the relations in the Mediterranean basin, where the source of legitimacy and good governance weren't the kings but the Roman state. As the label was still invested of prestige and programmatic undertones, its use accompanied the changes in western European politics.