Charlemagne and later \"Roman\" emperors

[Kai]

But I'm unaware of other Germanic kings being addressed as Augustus.

The Augustus-title was sometimes used on 'Germanic' kings (who actually had Roman citizenship, held Roman offices and commanded a Roman federate army). With regard to the use of it one must consider who is using it in what context. This is best demonstrated by Theoderic the Great.

We know of an inscription of a Roman senator hailing him Augustus. Indeed a large number of aristocrats obviously had an interest in accepting him, not the eastern Emperor, as highest suzerain. Conversely Theoderic quite openly toyed with this option.
In front of a clerical council he told the story of Aspar, a Magister Militum, who was offered the Augustus-title but declined. The message is simple enough: Theoderic could take the purple of the west if he wanted to, but he ostentatiously did not.
However a further change in tone can be found in official diplomatic correspondence when he never used any title but rex and his Roman offices, putting him unmistakably in second place behind the Emperor.

One also must be extremely careful in stating rex and augustus were used synonymously. It did happen but certainly not in the Ostrogothic kingdom and iirc generally not in the west. Clerical writers were the ones who used rex for the Roman Emperor rather frequently but this stems from the influence of the Greek language prevalent among clerics with Basileus being king as well as emperor. This had little influence on political language in the west and absolutely none at all on the legal discourses. More generally there is a difference between the use of rex for an augustus and vice versa