for its entire lifetime, the republic was highly oligarchic in nature. your goal is to make them fully, plausibly democratic - either in the Athenian vein or in a way unique to them. when could this be accomplished? what will be the result of a fully democratic Rome? could it be more prosperous or successful than OTL?
Actually, it's quite likely that the early Republic was actually pretty democratic. It's been a while since I studied this period, so I can't remember all the arguments pro and contra, but one piece of evidence I do remember is that the Senate's power was almost entirely built on its prestige, rather than on actual legal powers. So, it's likely that the popular assemblies were the main governing organs during the early Republic, and ended up being supplanted over the fourth century or so because Rome's increasing power required a more deliberative and permanent body to direct (especially foreign) policy.
Whilst Rome wasn't quite as democratic as, say, Athens -- even the popular assemblies did their voting in tribes, which favoured the rich and old -- it was nevertheless democratic during the first couple of centuries after the monarchy. If you want it to stay democratic, or become even more so, I'd say you'd probably need to keep it a city-state rather than a proper empire. Getting together the citizen body and saying "Hey, should we invade these guys?" works OK when the potential enemies are a threatening nearby tribe, but not nearly so well when they're a bit Hellenistic kingdom thousands of miles away, and the decision over whether or not to fight them requires a consideration of the logistical, financial, diplomatic, etc., issues likely to arise.