Why was "Caesar" borrowed for the imperial title in German rather than "Augustus"?

Caesar came first, and Caesar remained.

We observe that the first really big contact between Germanic tribes and Rome took place when Gaius Iulius Caesar was campaigning 'in Gaul'.

We observe also that under his adopted nephew Octavian the largest forays of Rome into Germanic lands was undertaken, bordering on colonising and provincialising Germany up to the Elbe river. The well known Battle of the Teutoburg Forest took place when Octavian was imperator, when we call him Augustus.

But what was his name then?

Name
As a consequence of Roman customs, society, and personal preference, Augustus (/ɔːˈɡʌstəs, əˈɡʌs-/; Classical Latin: [awˈɡʊstʊs]) was known by many names throughout his life:

• Gaius Octavius Thurinus (/ɒkˈteɪviəs/): He received his birth name, after his biological father, in 63 BC. "Gaius" was his praenomen, "Octavius" was his nomen, and "Thurinus" was his cognomen. Later, his rival Mark Antony used the name "Thurinus" as an insult, to which Augustus replied, surprised that "using his old name was thought to be an insult".54
• Gaius Julius Caesar: After he was adopted by Julius Caesar, he adopted Caesar's name in accordance with Roman naming conventions. While he dropped all references to the gens Octavia, people colloquially added the epithet Octavianus to his legal name, either to differentiate him from his adoptive father or to highlight his more modest origins. Modern historians refer to him using the anglicized form "Octavian" (/ɒkˈteɪviən/) between 44 BC and 27 BC.
• Gaius Julius Caesar Divi Filius: Two years after his adoption, he founded the Temple of Caesar additionally adding the title Divi Filius ("Son of the Divine") to his name in attempt to strengthen his political ties to Caesar's former soldiers, following the deification of Caesar.
• Imperator Caesar Divi Filius: From 38 BC, Octavian opted to use Imperator, the title by which troops hailed their leader after military success. His name is roughly translated as "Commander Caesar, Son of the Divine".
• Imperator Caesar Divi Filius Augustus: Following his 31 BC defeat of Mark Antony and Cleopatra, partly on his own insistence, the Roman Senate granted him the additional name, "Augustus", which he added to his previous names thereafter. Historians use this name to refer to him from 27 BC until his death in AD 14.

Augustus successor then renamed himself Tiberius Julius Caesar.

That means that from an almost equivalent to 'first contact' for the next 3 generations, the Germanic tribes when asking legionariries "who's your boss?" they would have heard at least some kind of "Caesar!".

In both cases, we also take note that the persons bearing that name would react quite peculiar when asked by anyone: "Are you a king?" Namely with "ONonononono, I am not a king, I am something different, the restorer of the res publicas!"

That it was around that time the name was borrowed into Germanic languages is evidenced by linguistics and language development. Caesar/Kaiser is one of the oldest Latin loanwords into Germanic languages. Loaning must have happened around that time that the mono-archicical biggest honcho of the Romans was always called some form of Caesar as the history of the sound of the German word proves: Shortly after this time the Latin language itself changed and the German loanword preserved the older sound:

Probably the oldest German loan word from Latin is Kaiser. The word goes back to the family name Caesar or its most famous bearer Gaius Iulius Caesar (100-44 B.C.) and was used as title under the following rulers. The word first came into Germanic and then further into German. The ahd. keisur (also -or, -ar) is probably also still used as title, but at the same time already as deonomastikon, as an appellative derived from the name with the meaning *ruler. Similarly, under the impression of an overpowering ruler, the Slavs later gave the name of Charlemagne, i.e. ahd. Karal / Karl, borrowed and used as ruler designation (Russian. король, Czech. král, Polish. król etc. for *King). But also to the Russian. царь *Zar is based on Caesar, it is a loan from got. kaisar, which goes back to Caesar via greek καῖσαρ . The sound laws allow here an approximate dating of the loan of Caesar into Germanic: It could not take place later than in the 2nd century, because until then the Diphthong ae was still spoken in Caesar, which also appears in ahd. keisur or got. kaisar; also the k- pronunciation was still preserved in the sound of Caesar, only in the 6th/7th century was k- palatalized, thus spoken as [ts].
_{–– Kurt Gärtner: "Lehnübersetzung und Lehnbedeutung vs. Lehnwort: Zu den Entlehnungen aus dem Lateinischen und Französischen in das mittelalterliche Deutsch", in Jens Braarvig & Markham J. Geller (Eds): "Studies in Multilingualism, Lingua Franca and Lingua Sacra", Edition Open Access: Berlin, 2018.}

Thus the oldest German loanword was well established when the Romans started to devise ever more complicated stylings for their 'emperor'. This more legalistic terminology of the Romans in Latin was then indeed adopted as well by the Germnic tribes that tried their luck at renovatio imperii. Charlemange as emperor was titled:

Karolus serenissimus Augustus a Deo coronatus magnus pacificus imperator Romanum gubernans imperium

That illustrates that Germans have really borrowed all available titles: Caesar, Imperator, Augustus.

While Augustus as a title is only used in Latin forms of titles and as a word declined in meaning and understandability, Caesar/Kaiser remained a very scarce given name. But August became common from the 14th century onward and besides also signifying a month it also took on a row of unfortunate connotations. Imperator remained a decidedly foreign word. Kaiser was the first, the most enduring, the most popular of these forms. And when a German speaking peasant would ask what the various long Latin titles (containing some form of Augustus) of Germanic emperors would come to mean, the answer would again be "Kaiser!"

That means that in medieval times the full title was a legal thing, and as such Latin dominated. Only the colloquial translation into the tongue of the people was still the old germanised word. That this got somewhat out of hand in being complicated is evidenced by the long-form of the titles of the kaisers of Austria. 'Long' as in three pages long.

When the German Empire was founded in 1871 the nationalist mood and complicated power politics within the Reich and in regard to Austria called for a simpler solution. A title that would still allow kings within Germany but also elevate the Prussian Hohenzollern king above the others. Thus the function of Federal President of the Reich got named with the word for 'biggest honcho in the empire' from 2000 years ago: Kaiser.