Pompey is probably not the guy to go the Sulla route. In fact, being known for his deliberate Alexander emulation, he's more likely than Caesar to actively seek some kind of kingly role (maybe not actually the title 'king', but something that effectively means the same). Playing a role there is the fact that he was the standard bearer of the optimates, while Caesar championed the populares. These groups were more symbolic in what they truly represented than anything (i.e. the optimates didn't shun populism at all, and leading populares were all elite figures rather than 'men of the people') but Pompey's more aristocratic support base would allow for a more monarchial/aristocratic style of governance.
So, really, your best bet is for that to happen, and for it to cause so much resentment that Pompey ends up killed for it. But unlike in OTL, the liberatores (or their analogues, whatever they call themselves) enjoy far more support, and get to restore and reform the republic.
I don't think this is actually the most likely scenerio, as Pompey was too acutely aware of political realities to turn so many people into enemies. More likely, his existing rivals use accusations of monarchism to set alt-liberatores agaist him, and then quickly do away with those liberatores. After that, Pompey's rivals duke it out as Caesar's would-be heirs did in OTL, until one comes out on top and claims absolute power. Basically, I'm conviced that the Republic had been descending into an intensifying rivalry for such power for some time, and that by the time of Caesar, it was only a matter who would gain that power. "Saving the Republic" in the sense of bringing it back to anything other than leadership by an autocrat is highly unlikely that late in the game. Nevertheless, attributing the foolishness of overconfidence to Pompey and going the route I described above is probably the best way to make it happen.