I am not aware of any real concerted effort that could be considered a plot or even an idea to restore the machinations of the state from the Imperial administration back completely to the Senate, or "republic". Let me explain why no one probably considered it a good idea. The transition of power from republic to empire was very very gradual. When Octavian (or Augustus) took over as the first emperor he did not call himself emperor but instead "first citizen" or something along those lines. He did not admit he was the emperor as the republic was not only a form of government but also a sort or religion to a certain segment of the Roman population. He could not simply come to power and declare himself absolute ruler. Its hard to understand today but the republic was more of an idea/a tradition than a concrete form of government as we know today. It was instead just "the way things were done." There were certain rules that everyone knew that were passed down from generation to generation. The Romans had no written political documents such as our constitution or bill of rights. The republic was merely a tradition, a culture really, that was passed down from generation to generation and instilled in young Romans since birth. It however only really applied to the elites and family's of the ultra wealthy. Regular Romans had no representation in a real sense and would just vote for whoever their tribal leaders told them to, tribal leaders who were largely controlled by the elites previously mentioned. This system worked well until Rome expanded very very quickly in a very short period of time. This system began to break down as the Empire grew and you had more citizens from outside of Rome and Italy. You had massive influxes of slaves that took the jobs of freemen. You had the consolidation of land by a handful of wealthy powerful Romans. You had new generals and entire peoples who wanted to ascend the political ladder and be treated as citizens or elected to higher office but were not seen as part of the traditional Roman elite that were allowed the same freedoms and rights as those at the top. Enter Julius Caesar a politician who grew up in a time of great political upheaval- multiple civil wars, Roman generals fighting each other, the massive Spartacus slave rebellion, food shortages, massive unemployment due to elites buying slave labor, the destruction of small family farms in favor of massive industrial farms etc etc etc. Caesar whether a man of the people or an ambitious tyrant had utilized the weaknesses of the republic to garner support of the general population. Caesar's power flowed from decades of failures of the Senate. When Julius Caesar became the first unnamed emperor he did so because it was only inevitable in his eyes. Rome could no longer survive as a republic because she no longer was one. She had to become an empire in order to govern like an empire. The republic, controlled by a handful of elites in Rome, who's families had been in control for hundreds of years, was incapable of caring about or managing an empire with multiple cultures, peoples, territory's, and needs. Caesar famously once said "the Republic is nothing, merely a name without body or shape." He was right, the Republic was basically all of the rich people in society getting together and agreeing that they were the ones in charge, that their needs would be met first, and that they would play fairly among themselves and hold elections between themselves. The moment that anyone played ball with any minority groups or wanted to establish a more fair and effective form of government they were excluded from the "republic" and deemed an enemy of the state and killed. It was a wholly ineffective, corrupt, and repressive from of government that only benefited a few hundred, maybe a few thousand people. It was absolutely nothing like the republic we have today ; ). It was more like an oligarchy but i would hesitate to put any label on "the republic" as it was a completely Roman unique form of government that is hard to understand in modern times. The republic turned into empire out of necessity, not because someone wanted to take power and not relinquish it. The Roman republic was incapable of governing a world wide empire, with all of the infighting, ambitious young nobles vying for prestige and titles and honors. It was simply a totally antiquated form of government.

But lets say that it was a great from of government and it totally rocked. When Caesar died and his heir Augustus took over after anotherrrr massive civil war, he still let the senate do most of the governing as long as no one did anything against his wishes. It was only later through decades and then centuries of time that the emperors developed into absolute rulers like we know them and we are familiar with today. Eventually the emperors would start hand picking the senators he wanted directly into positions and hold fake elections. Then the emperor just took over the administration of the most valuable and politically sensitive territories himself. The "breadbasket of Rome" Egypt was the most famous and important of the emperor's personal land. He could not possibly let a senator govern it for him because then he could potentially hold the food supply of the entire empire hostage. These are the intricacies that made the republic impossible once the empire came into being. No one man, other than the emperor, could be trusted to not become overly ambitious and decide to mount a coop while administrating a province like Egypt. This is why through time the Senate just became a body of the most influential and wealthy people of Rome who's role was mostly ceremonial. Again, this transformation took effect over hundreds of years. Once the empire had been fully and totally established the republic was nothing more than a forgotten dream. More like a nightmare given how the republic ended, with almost a half century of brutal war and political killings and major civil strife. Without any legal/political documents to structure what the Republic was or how it operated people a hundred or two hundred years after its collapse probably had no real idea. But more importantly who would get to decide who could and couldn't participate in it? And what was preventing another Caesar from destroying it once it had been reestablished? Who was going to be willing to put their life on the line and overthrow the Emperor to re establish a form of government that had totally failed?

Edit: Forgot to put sources y'all

Adrian Goldsworthy- Augustus

Caesar- Matthias Gelzer

Antony and Cleopatra- Adrian Goldsworthy

Caesar- Adrian Goldsworthy

How Rome Fell- Adrian Goldsworthy

The Twelve Caesars- Michael Grant

The Assassination of Julius Caesar- Michael Parenti