There is no way that Genoa reconquers Corsica on its own in the 1760s. They had been trying and failing for 30+ years at that point, and the only success they had ever achieved was when a foreign army was intervening on their behalf (and that only ever lasted as long as the foreign army was willing to stick around). They simply did not have the military power or the resources. That's not to say that an end to the rebellion under Genoese rule isn't possible, only that it could not be achieved with purely military means.
IMO the last possible POD for a peaceful Genoese restoration in Corsica is 1757, when Paoli's men killed Mario Emmanuele Matra. Matra was Paoli's chief rival for power in the national movement, but nevertheless had sought the aid of the Genoese to fight Paoli's supporters. Matra came very close to killing Paoli in 1757, and if he had done so, it's possible that he could "win" the civil war. Even then, I don't think a reconciliation is likely because once Paoli is dead Matra doesn't need the Genoese anymore, but it's at least plausible that a Matra victory in 1757 might make the national movement less militant and independence-oriented such that Corsica eventually, somehow or other, returns to the Genoese fold.
As for Napoleon's altered future, one must keep in mind that while Genoa may have been a republic, it was by no means a meritocracy. Genoa was an aristocratic republic with an emphasis on the "aristocratic" part. You tend to see the same family names rotating in and out of the Dogeship and senatorial positions, because leadership was confined to a handful of old Genoese aristocratic families who generally did not let outsiders into their club. The Buonaparte family was important locally, regularly holding civic elder positions in Ajaccio, but that's a very far cry from power in Genoa itself. That they were "noble" matters very little. Undoubtedly young Napoleon could have enjoyed some moderately high-status career like being a doctor or lawyer or something, but the military would have been unattractive, as the Genoese military was a sad little affair in which most officers were Genoese (read: not Corsican) noblemen and foreign mercenaries. There were a few Corsican battalions but Corsicans were discriminated against in the officer corps and tended to have their careers ended early.
That said, however, Corsica had a long tradition of "exporting" soldiers to other states, especially Venice but also the Papal States, Naples, France, and Sardinia. If you were an ambitious Corsican who wanted an officer's career you didn't work for Genoa, you went abroad. If Napoleon is born in a Genoese Corsica, and if he still ends up in a military career, then the most likely outcome is that he ends up as a Venetian or Neapolitan captain or something. The best outcome he could probably hope for would be to get into the French Regiment Royal-Corse, which was actually a military unit that did things, but then you've put Napoleon back in France and presumably that's not what you're looking for in this thread. Elsewhere, and in Venice especially, there's not all that much of an opportunity to win military glory, nor for a foreign soldier to attain any kind of political power.