Make a democratic movement emerge in Rome among the Populares and make it strong enough to reform the Roman aristocracy not into a monarchy as OTL, but into a democracy based on the model of Ancient Athens.
My proposition is basically to get the Gracchi in power and to have them give more power to the equites (right to take part in jurys for example).
Once this Roman middle class is in power, politics radicalize even more and democratic reforms are viable.
They did get the equites on the juries. Gaius Gracchus successfully passed that legislation in 123 BC. Only equites could be jurors. Turned out the equites were very corrupt, since they always favor the publicani or tax farmers, since tax farmers were equites. Thus, things get even worst for provincials!
What about way after the republic?So, how can we have a democratic Rome (AFTER the victory over Carthage, since a democratic Rome as a minor power is quite senseless in my eyes).
What about way after the republic?
The post-Alexandrian Hellenistic trend of small- to medium-sized oligarchies turning into large monarchies - which influenced not only Rome, but also, for example, India - was powerful and not quite easy to stop.
But, as the Romans experimented with their Augusti beginning with Octavian, they might have seen the flaws of concentrating power in one person more clearly once again. One mad emperor certainly wasn`t enough, but after various weak ones and usurpations and all the like,
I indeed ask myself why no political philosophy emerged which denounced the Principate as dysfunctional and, looking back to various classical models, proposed broader participation of the populace.
Thing is, they`d have to develop a concept of "federalism" at the same time. It existed, in the various Greek "koina", but never quite as stable as the Roman imperial alternative.