After it became an empire, could it have went back to its Republican roots? When would be the best time to do this?
The turning point is surely the accession of Claudius - there is the last opportunity to truly restore the republic (as a fact, not as a form - formally, the empire was a republic until... 1453). After 41 AD, a restauration is very very unlikely. The ruling, senatorial class has aranged with the new state and the emperor can rely on a strong but dependent order of knight.
The question is now: there is the end? If the Roman empire crumbles in 476 and 1453, as OTL, you'll certainly never see a new republic. But if you decide to prolong the empire's existence (as a strong and united block), technological, economical and social progress will maybe form a new middle and ruling class (e.g. merchants trading with India and the Americas and later factory and mine owners) wanting to take part in the state's administration and legislature. If the emperor (or the court if you some Japanese-like shogunate governing the empire instead of the empire) is cooperative, it might end up in a constitutional monarchy as OTL Britain or Belgium, slowly evolving into a popular state with universal suffrage - but if the old elites are obstructing the process (like in France), radical elements can grew in power, overthrow the emperor and create a new republic.
The old republican constitution did simply not work for an mepire of this size. And the romans knew that. A more constitutional monarchy with increased power of the senate might have been possible. As earlier you start, the better. Latest with Vespasian, who was the first emperor, who got one integrated imperium containing all the different rights of an princeps, I see no chnace anymore.
I think that it is very likely that the imperium of the Tiberius, Caliguka, Claudius etc. was also granted by a lex de imperio. These laws just aren't preserved until today.