I've been doing a little research and have a few thoughts on the Republic's major problems.
1. Political Violence
Far as I can tell, this was extremely limited until the assassination of Gaius Gracchus, but from there it escalated, to Sulla's, Cinna's, and Sulla's second capture of the City - previously, a Roman army attacking the city was unheard of, other than by treason to a foreign enemy. While this violence may have originated in the debate over the latifundia (see below) and the socii, it clearly did not stop there, and only ended with Octavian as Emperor. But even that is misleading - the Empire frequently (starting with the Year of the Four Emperors) saw brutal civil wars.
So an ideal Republic needs to stop mob violence, to the extent possible, and CERTAINLY the idea of a Roman army marching against the city.
2. Death of Small Farms
Rome before the Punic Wars put an astounding number of men into the field for its population - probably not matched again until the French Revolution. These armies were based on the propertied plebs, equites, and patricians, and was a patriotic, loyal, and highly motivated citizen militia. But Roman soldiers, starting with the Second Punic War and the acquisition of Hispania, began to be posted abroad for long periods of time (5-10 years). Many small farms failed and were bought up by the wealthy patricians, forming the vast landholdings called latifundia. This was a sort of vicious cycle and the number of eligible soldiers, based on property requirements - a soldier had to equip himself, went into a death spiral.
Marius, attempting to outwit Sulla's appointed command during the Jugurthine War in Numidia, was able to drastically lower the property requirements and equip a professional army overwhelmingly composed of the very poor. This army was generally loyal to the general, not the Republic, and unfortunately quickly became the norm in the First Century BCE.
So an ideal Republic might address some of the terminal problems by saving the small farms. A problem: Romans of the era were very aware of this problem. But there were major interests invested in the latifundia. Furthermore, any enterprising magistrate (either of the Gracchi or Marius, say) who could solve this issue would become too powerful. The opposing senators would rather stop him than solve the issue and let him take all the credit.
3. Lack of a Chief Executive
Other than occasional dictatorial terms in a crisis - these were to last 6 months - Rome had no chief executive between Tarquin and Sulla (a 400 year gap). Instead, two consuls, four praetors, and often pro-magistrates (extended term consuls and praetors usually posted to the provinces) and, later, tribunes and aediles of the plebs, executed policy as best they saw fit. This included military, legislative, and business issues, and while it was meant to serve the body politic it primarily served the interests of the individual and to some extent his gens. There were no parties, official or otherwise; there were certainly political alliances but these were to be breached when they had run their course. This explains the apparent selfishness of all the major actors in the First Century BCE - selfishness was what was right and expected (though in the service of the SPQR).
The system worked surprisingly well until the end of the Second Century BCE, but then began to break down as ambitious politician-generals served no higher authority but themselves.
Since I started this thread to find a better alternative than an Empire, why do I want a chief executive? Because, while elected executives might be inept, malign, or otherwise unfortunate, they can be replaced without bloodshed. In the Empire, only death (until much later in the ERE, and with very rare exceptions like Diocletian) could replace an Emperor.
An elected executive with a term from 1-6 years, checked by the Senate and other magistrates, is a total reimagining of the Republic. But could the original system of the divided magistracy survive long term, if the other problems are solved? Other problems would surely arise in time, and, unfortunately, the 2nd and 1st Centuries BCE showed that the system was poorly equipped to decide major issues that required innovative policies.
4. Lack of a Formal Constitution
Rome's constitution is often compared to Britain's, in that it was unwritten but focused on precedent and very real and binding. But British constitutionalism has been in the direction of a more popular (in both senses) government, with a few hiccoughs, for centuries. In Rome, the adaptability of precedents was very useful for a number of years, but then ceased to be. Mob violence, Marius circumventing Sulla's consular authority, Sulla's marching on the city, Sulla's proscriptions - to an extent, all of these became precedents. Even Caesar's crossing the Rubicon and Octavian's establishment of the Principate could be understood as a new interpretation within the exisiting law.
Unwritten constitutions, as modern Britain and Israel prove, can work; but Rome's did not. An ideal Republic likely requires either a formal document or a much more official, well-established, and comprehensive body of precedent. Sulla came the closest to enforcing a conservative version of the Republic, but many of his reforms failed to last.
5. Management of Provinces
Until the aftermath of the Social War few Italians even had citizenship; so it is hardly surprising that the provincials were on an unequal playing field in Roman law. Governors frequently abused their position to become rich; the provincials could only legally challenge them by, at often great expense, sending a deputation to Rome; and these challenges usually failed.
Extending citizenship and including the provincials in the Republic is part of the long transition of Rome from a city which happened to have acquired an empire into a vast state; and it does not have to be solved as quickly as the other issues to save the Republic. However, it must be addressed at some point if a large Republic is to survive.
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With all this in mind, here are a couple of POD ideas:
A. The Gracchi outwit their rivals, destroy the latifundia, and extend citizenship to the socii. An early Civil War leads to a victory for the socii, plebs, and pro-Gracchi patricians over the conservatives and large landholders. In the aftermath of the victory it becomes necessary to further reform the Republic.
B. Sulla cements and formalizes his reforms by promoting a *constitution*. Sulla lives longer, is more politically astute, and doesn't retire as early. While Sulla packs the system with his supporters, the structure he establishes is sound and long survives his personal rule.
C. Pompey wins the Civil war, the Liberatores assassinate Antony as well, or Octavian and Antony are killed by the Liberatores. I'm unsure that any of these variants are soon enough to save the Republic, but undoubtedly Caesar by his successes put it on the immediate road to... something else, in OTL the Principate. Any of this defeats for Caesar and his heirs stops the immediate cause, but not the underlying causes, for the end of the Republic. So this set of PODs requires the additional note that afterwards, Pompey/Cicero/Brutus/someone decideds to drastically reform the Republic.