Interesting question!

The iconography and epithets attached to the Pharaoh were applied to Augustus in Egypt. However, Augustus never made a concerted effort to present himself as a successor to the Egyptian dynasties which had come before, and the Roman conquest was quite plainly shown to be the beginning of a new mode of rule in Egypt.

Prelude

Before I begin talking about royal ideology in the Roman period, I want to establish context by exploring the idiosyncrasies of Ptolemaic kingship and dynastic iconography. This is relevant because it was the Ptolemaic dynasty which established the recent precedent of a Pharaoh who was also a foreign ruler. Rulers like Cleopatra had a dual role as Hellenistic queen or king on the one hand, and Egyptian Pharaoh on the other. Egyptian perceptions of the Pharaoh had already been forced to adapt, in the 1st Millenium BCE as foreign dynasties repeatedly conquered Egypt in whole or in part.

There was a concerted effort by Alexander to establish himself as a legitimate Pharaoh, including a supposed visit to an oracle of Amun that granted him the alleged approval of what was one of the principal deities as well as a later story (which was incorporated into the Alexander Romance) that he was actually the son of Nectanebo II. Ptolemy Lagos continued in this tradition and presented his dynasty as liberators of Egypt for propagandistic purposes, which was further strengthened by Ptolemy II with the establishment of the dynastic cult which placed the Ptolemaic kings into a divine capacity that both Greeks and Egyptians could accept.

In Hellenistic contexts such as the Greek-dominated poleis as well as the Asiatic and Aegean holdings of the Ptolemaic kingdom, the king is clearly defined in his role as king, albeit in a sacred capacity typical of the Hellenistic era, and artistic conventions remain strikingly Hellenistic, even on coins distributed in Egypt. Meanwhile in Egyptian contexts, such as temple dedications and monuments, they are portrayed as Pharaohs in a very traditional sense.

This dual identity developed organically to allow the Ptolemaic rulers to play to their audience, assuring the mutual loyalty of the Greek and Egyptian populations.

Far from accepting the Ptolemids as an Egyptian dynasty and passively submitting to Macedonian conquest, the Egyptians were plainly aware that they had been annexed by a foreign power. This is reflected not just in cases of insurrection, but also in literary and artistic traditions. For instance, the 1st Century BCE funerary stelae of Pasherenptah and his wife Taimhotep, Ptolemy XII is mentioned extensively and reference is made to

the Residence of the Aegean kings, which is on the shore of the sea on the west side of the 'i/J-district, the name of which is Raqote [meaning Alexandria]

This line is interesting because it not only establishes distance between Egypt and Alexandria, but refers to the Lagides as the "Aegean kings", denoting their foreign origin in an apparently non-hostile context. Examples like these, although fairly scarce do demonstrate that the idea of a foreign Pharaoh or of an increasingly malleable and symbolic role of Pharaoh, was developing.

Pharaoh, Augustus & Prefect

For the most part it would seem that Augustus made it quite clear that he had no desire to be king of Egypt, in part because of the anti-king sentiments you note. Augustus was well aware that the Roman fear of autocracy was part of what had brought down his rivals Antony & Cleopatra, as well as his own uncle Julius Caesar and he had no intention of repeating their mistakes. He was not crowned at Memphis and Diodorus Siculus states that he refused to pay homage to the sacred Apis bull or to the mummified kings, both of which had been one of the principal cults patronised by the Ptolemies.

This attitude did not mean that Augustus intended to do away with the traditional Egyptian cults, they just now lacked the direct involvement of Egypt's absolute ruler although he continued to be invoked in dedications. The Emperor's presence and dedications was likened to that of the Pharaoh in his ability to encourage fertility and flooding of the Nile. The imperial prefect, as an extension of the Emperor's will in Egypt, oversaw offerings and libations to the deified Elephantine, source of the Nile, in his name. It appears that the prefect fulfilled most of the day to day ritual functions of the Egyptian king in place of the Pharaoh. This unclear relationship between perfect, Emperor and crown led to the fall of Cornelius Gallus for example, after Augustus came to suspect him of having kingly aspirations. It is no surprise that such a position would breed jealousy, anyone asked to oversee one of the most wealthy and (initially unruly) provinces in the Empire would only provoke suspicions.

The state sponsored dynastic cult of the Ptolemies was done away with, and replaced with the Kaisareia. It is difficult to tell to what extent the living Emperors were considered deities in Egypt or whether they were merely considered divine and deified in death. Although their likenesses were enshrined and carried in processions next to that of deities they were often depicted as offering to the gods and are invoked in an intermediary capacity.

Spiritually, the Pharaoh served as the intermediary between the people and the gods, and promoting balance in the Egyptian world order of ma'at. This spiritual aspect of the Pharaoh was somewhat altered in the Ptolemaic period, as temple inscriptions occasionally make reference to Osiris in place of the Lagide Pharaoh whose cartouches were left blank to signify quasi-legitimacy, and Ptolemaic monarchs began delegating more religious duties to priests and intermediaries.

At the same time Pharaonic statues were considered to hold the divine presence of the Emperor and it was for this reason that they could be appealed to by those in dire need, and these statues were sacrificed to and received cult. Beyond that, they are never referred to as gods outright, but they retain epithets like Soter (Saviour) and Eurgetes (the Benefactor) which may be intended to carry divine or semi-divine connotations but may also merely praise them as rulers. Whatever the case, while they were not a direct continuation of the Ptolemaic cults by any stretch of the imagination, the Imperial cult filled a similar role as priests were generally also local or regional officials and, offerings and libations were given their on the Egyptian New Year.

With the lack of a Pharaoh, or at least an absentee Pharaoh, the spiritual significance of the king became transferred to a a symbolic stand-in for local religious contexts. There are a plethora of temple images dating to the reign of Augustus that depict him as Pharaoh and feature blank cartouches which suggest that he was not seen as fully legitimate. In some temple reliefs and stelae, his cartouche bears the Greek title often attributed to him of Autokrator Kaisar. The most famous possibly being the images at the Kalabsha temple in Lower Nubia which was at the time a part of formerly Ptolemaic Egypt. These reliefs bear the name of Augustus and unmistakably portray him as an Egyptian Pharaoh offering gifts to the gods and being consecrated as Pharaoh.

The decision to portray him in this fashion can not be certainly attributed either to Augustus or to the priests. Given that Augustus carried out many projects to construct, renovate and expand temples in his early reign, it is possible that he was depicted in this fashion as part of a propaganda campaign to grant legitimacy to his rule and appeal to tradition. On the other hand, it is possible that this decision was quite often made on the part of the priesthoods, who felt the need for some form of a Pharaoh to uphold ma'at.

Many depictions and references to Augustus as Pharaoh exist, not all of them even from temple contexts. Some bear the usual titles in Latin or Greek but the Egyptian inscriptions reference him as Pharaoh. These examples make it likely that many Egyptians considered him to be a sort of facsimile of a Pharaoh. This carefully cultivated image meshes well with the epigraphic evidence of prayers being offered in the name of Augustus, and the continued role of the Emperor in ensuring the fertility and security of Egypt. Overall, there seems to have been a perception of the Emperor as filling the capacity of Pharaoh but not being defined by it the way the Ptolemaic kings had been, as he existed as master of a Roman world which included Egypt but was not centred around it, marking perhaps the greatest ideological departure of Roman rule.

The practice of associating the Emperor with Pharaoh did not end with Augustus, and continued into the reign of Diocletian. Representations and descriptions of the Emperor as Pharaoh making offerings to the gods were especially common, indicating the importance of the Pharaoh's spiritual role in the Egyptian mind. Hadrian actually paid great patronage to the Egyptian cults, constructing temples in Alexandria, Canopus and Memphis, and mediated the dispute between Alexandria and Memphis as to what city would hold the new Apis Bull. Hadrian also had his lover Antinoous deified and associated with the local Osiris cult following his death in Egypt.

The advent of Christianity heavily contributed to the demise of overt references to the Emperor as Pharaoh. The divine nature of Pharaonic kingship, like so many aspects of traditional Egyptian culture, was irreconcilably pagan and did not last the state's efforts to expunge heretical and polytheistic traditions from Egypt.

I always enjoy reading your detailed and lucid answers. Thanks.