You Need a Better Roman Constitution
I've been studying this question, and I say persistence til today's not completely impossible (it's been done a few times here), but not so easy, either. You'd have to find a way to get to a better ending constitution than the Roman Empire had, which was a covert absolute monarchy. You need at least one or two checks and balances.
After all, in OTL, Hero of Alexandria's inventions of steam gadgetry and binary state tape went unfollowed-up for a millenia and a half because the Empire was the Empire. Such a better gummint would probably still see developments somewhat slower than OTL post-Renaissance because Rome was a strictly military specialist.
The Roman REPUBLIC was unusually steadily successful at war for century after century. In its enterpreneurial and innovative Republican phase, freedom let war-specialized innovation brought Rome to the lead in miltech and kept there for century after century. Elections chose generally good leadership. The Republic was a wank personified.
Under the Caesars' monarchic rule, all those advantages went away. The man who failed to conquer Germany was a chosen more for his buddyhood to Caesar than his ability to win a vote. By a century or two later, their neighbors had caught up with them in miltech; Roman turf went from quickly growing to slowly decaying; though the borders moved around and even very temporarily grew under Justinian, they overall shrank to nothing. Roman society similarly decayed, growing more static with the centuries. By its fall, it had grown so Talibanesque, its fall was, IMHO, truly a gain to the world.
It ended with their second capital city, Constantinople, being taken by the Turks by better technology - cannon. The cannon were developed by a Christian engineer whom had come to save the city. But the emperor and his court lacked the basic wisdom to value an inventor and treat the cannonmaker decently, and off he went in fury to the much more reasonable Ottomans, and it was bye-bye Roman Empire. That was quite the opposite reaction the high-tech Roman Republic would've had.
So, it can't be done with the kind of Roman Empire we had. You at least need some checks on the Emperor's power, a constitutional monarchy like Rome started with. Better still, IMHO, is to have the much-better, but vulnerable in Caesar's day, Republic-style constitution survive, especially if you want it expand much. The Republic was amended unwisely to allow a more suitable number of men to serve by one Marius, opening up a hole for warlordism (Sulla). Julius Caesar conquered the Republic out of its warlord misery, whille nephew Octavian turned it to absolute monarchy with Republic-like fronting. The Caesars are probably bad material to work with to get a better Rome, since they OTL liked absolute power just fine.
IMHO, it gets pretty improbable to have the constitutional change, whatever it is, happen much after a century after the Octavian Caesar started the Emperor's absolute rule. Back then, there was a feeling that Romans were worse off than under the Republic, but nobody saw a way to get back. Later, people saw the absolute Empire as natural, and the interest in the Republic vanished.
Here are
some nice, long threads
on how the well=checked Roman Republic fared after it went to
unchecked monarchy. Enjoy!