Well, a Roman concept of the Mandate of Heaven would be immensely useful and would permit a way to *have* dynastic changes and even civil wars that produce them that are less devastating to the overall system.
Here would be my proposal.
Caesar isn't assasinated. But what his enemies' fear is true--he is becoming increasingly unstable and power hungry. At the same time, he is as charismatic, ruthless, and talented as ever. Result: he has himself crowned king, engages in adventurism to the East, massively reorganizes the Roman state, and slaughters a whole bunch of his enemies. He is eventuall assasinated some 15 years later.
His reforms don't really stick but they do succeed in destroying much of the prior societal basis. There is a big period of chaos and another civil war. Augustus again comes out on top, but not as much as a supposed restorer of the Republic. His claim to power is as successor to the divine Ceasar. He does recreate some Republican institutions and Roman rule, its not pure autocracy, but his basis for rule is neither the Republic nor his own merit, its descent from the God Julius.
After a few generation of rule by the Julians, the machinery of state is breaking down, there's another civil war, and some other ruthless soldier-statesman comes to power. But after him the state is again ruled by primogeniture on the Julian model, just with his descendants instead of the Julians. The Roman theory here is that the divine blood is diluted after a while, as evidenced by disorders and ineffectual rulership, and so a new demi-god is ordained by the deified protectors of Rome to establish a new dynasty.
Based on hints from Plato, maybe the educated classes suspect that the new demi-god is actually a reincarnated Caesar. In time, you might come to have some kind of state religion where Boddhisavata types give up their divinity to be reincarnated to rescue the state of Rome.