Prelude: The Aegean Kings
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. On the other hand, Egyptian contexts, such as temple dedications and monuments, portray them as Pharaohs in a very traditional sense so even at this early stage we can see ideological dissonance in the ruler’s role in various contexts. However even this seemingly simple duality is further complicated by the meeting of these two traditions both in stylistic traditions and in subject matter, for instance the portrayal of Ptolemy V on horseback smiting his enemies on the Raphia Decree stela is only superficially Egyptian in style as the Pharaoh is not depicted on horseback in earlier portrayals, and the use of Egyptian iconography in otherwise predominantly Hellenized religious contexts also shows a degree of compromise. This system of cultural exchange was likely more organic than intentional, and I would argue the Hellenistic kingship and ruler cult, and the Pharaonic role as deity were never the same, but merely cooperated to assure the mutual loyalty of the Greek and Egyptian populations.

The second point is that the Egyptian perception of the Pharaoh had already been forced to adapt, in the 1st Millenium BCE on the one hand as foreign dynasties repeatedly conquered Egypt in whole or in part, but following the conquest of Alexander in particular. 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 ( nw) of the Aegean ( 3w-nbw) ‘”kings 1 , which is on the shore of the sea (w3 -wr) on the west side of the ‘i/J-district, the name of which is Raqote.
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.1 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. Despite all this however, 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. This royal cult which likened the king and queen to Osiris and Isis is superficially similar to the Roman Imperial cult and the divine role of Hellenistic kings did influence it to some extent but it was still not a direct continuation of this by any means. Moreover, although the Egyptian temples carried out legal and administrative tasks much like Roman temples, it should be noted that the Roman Imperial cult was both secular and sacral.
The practical role of the Pharaoh, Augustus & the Prefect
You bring up a good point in asking about whether or not Augustus filled the more traditional functions and ceremonies of Pharaoh and whether or not some of these duties were relegated to the prefect. For the most part it would seem that Augustus made it quite clear that he had no desire to be king of Egypt, 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. It was also during his reign that Egyptian cults were prohibited from the city of Rome, in large part due to their association with his vanquished foes Marc Antony and Cleopatra. This attitude did not mean that Augustus intended to do away with the traditional Egyptian cults, on the contrary they remained as popular as ever, only now they lacked the direct patronage of Egypt’s absolute ruler who 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, and the prefect oversaw offerings and libations to the deified Elephantine, source of the Nile, in his name, and indeed, fulfilled most of the day to day ritual functions of the Egyptian king.

The state sponsored dynastic cult of the Ptolemies was done away with, and replaced with the Kaisareia which were centrally managed on a nome level. It is difficult to tell to what extent the living Emperors were considered deities in Egypt or whether they were considered divine and deified in death which was an important distinction. 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. At the same time their 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 Ptolemaic epithets like Soter and Eurgetes, 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.
The Imperial Cult & Pharaonic Ideology
For the most part, the Imperial cult in Egypt took on a distinctly Hellenistic form where it was dedicated by Graeco-Roman locals who wished to ingratiate themselves with the Imperial family. As a result of both the cultural affiliation of those dedicating it and the degree of Romanisation in most cultic images and statuary means that Egyptian influences only appear in oblique, syncretized fashions that build on traditions which were by then well established in the Hellenistic and Roman worlds. In this sense, associations with more familiar deities like Apollo and Jupiter/Zeus would have been interpreted by an Egyptian audience as Horus and Amun. Imperial cult centres were built with distinctly Roman architecture, and although references to the Roman state or the goddess Roma are conspicuously missing in Egypt, this could be due to an increased prominence of the person of the Emperor and his family which would have been more in keeping with traditional ideas of dynastic worship, reconciling past cultic tradition with new, foreign symbols.

That said, as I mentioned above, the Pharaoh was also important on a religious level, as he 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 (which the Ptolemies who had no knowledge of Egyptian would never find out), and Ptolemaic monarchs began delegating more religious duties to priests and intermediaries. Inscriptions and reliefs continued to depict the Pharaoh as intermediary between the gods and semi-divine and the Ptolemids were careful to provide patronage to this aspect of their association, with it being notably forefront in the propaganda of all the Ptolemies but particularly of importance to Ptolemy V, Ptolemy XII and Cleopatra VII who took a more personal hand in these traditional functions once more and other members of the dynasty similarly may have been more or less connected to this aspect of rule. The title of a priest attested to under Augustus as “Prophet of Caesar” seems to belong to a priest of Ptah which was connected closely with the Ptolemaic dynastic cult, and the title itself is probably analogous to “Prophet of Pharaoh” which is attested to in the reigns of Ptolemy XII and Cleopatra VII.
In the Roman period, the temples lost a great deal of their administrative and political power but remained the centres of worship, 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 is 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, however in other 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. In any case, 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.

This second interpretation seems most likely given how many depictions and references to Augustus as Pharaoh exist, not all of them even from temple contexts, with some bearing the usual titles in Latin or Greek but the Egyptian inscriptions referencing 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. In particular, 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 mediating the dispute between Alexandria and Memphis as to what city would hold the new Apis Bull, as well as having his lover Antinoous deified and associated with the local Osiris following his death in Egypt.
Sources:
Ch. 3 Egyptian Elite Self presentation in Ancient Alexandria: Between Egypt and Greece by W.V. Harris and Giovanni Ruffini

Pharaoh, Basileus, and Imperator: The Roman Imperial Cult in Egypt by Gregory Steven Dundas

Art of Empire: The Roman Frescoes and Imperial Cult Chamber in Luxor Temple by Michael Jones and Susanna Fadden