IIRC, just after the death of Caesar, Antony manages to intimidate the tyrancides and get them to flee Rome. The second triumvirate then forms to punish the fled assassins.
The easiest change I can imagine is that Cicero grows a backbone. He is neither a Caesarian Populares nor an ally of the tyrancides. Let's say he decides to join Brutus and Marc Antony in giving a eulogy of Caesar, and he does a very good job of it. He now has a genuine following separate from either faction.
The most immediate issue is to decide the character of the murder. Brutus and his allies have not yet been declared tyranicides because of all the complications decarling Caesar a tyrant would have implied. Antony still wants to kill the murders and Octavian still wants to safe-guard his inheritance. So things go pretty much as OTL until the Lex Titia is proposed. Cicero intervenes before it can be brought to a full vote. He begins neogtiations to uncouple the alliance of Antony and Octavian: Lepidus will be elected Consul for the coming year, with Cicero as his colleague. Caesar's will will be ratified by the Senate and Antony will be given full command of legions to apprehend the fled Senators who have begun to raise an illegal army of their own. As the war continues, though Antony still hates Cicero and vice-versa, Octavian counter both and TTL's informal second triumvirate become Cicero-Octavian-Antony. I believe that this is in line with Cicero's efforts to play Antony and Octavian off one another, but here he is more successful.
If you're in the mood to really save the Republic, then at some point Octavian, still the junior partner, propose to Cicero that to preserve the Republic changes must be made to ensure that a situation like that of Caesar's rise (to Cicero) and his death (to Octavian and Antony) doesn't again threaten the power of SPQR. They will need to fix two problems, IMHO: the personal loyalty of troops to their commanders, rather than to SPQR, and the need for a permanent executive authority to perserve order at home and abroad.
I'm not sure how they would solve the problem, but I'd feel confident that they work something out. The solution I've been mulling over is the creation of an additional office of Consul (now there will be three) and that each consul will serve a term of six years, with one being elected every two years. The three Consuls constitute a "Consulate" and certain decisions require the consent of 2/3 others 3/3. As to the army, I think the issue there is a more general reform of the popular assemblies to at least legalize the role of the army: perhaps a "Comita Militum"? My thought is that it's fairly impossible to curb the loyalty of troops to a commander, but if you enfranchise and empower all of the troops at once, then perhaps you re-create plebian-patrician politics in the army and thus stimy the effect. The position of Dictator is abolished in law. In an emergency (which must be recongized by a vote of the Senate and a unanimous ruling of the Consulate), the most recently elected Consul will assume temporary veto other the dissent of his colleagues.
Alternatively, the three-Consul situation could be the comprimise that emerges from negotiations over the Lex Titia rather than a later development. I have no idea how plausible this nor would I consider myself as knowledgeable on the period as others seem to be. I bow to their wisdom on the topic.