This is quite a challenge.
I think that what we need is for Sardinia to fail in its quest to unify Italy, say that for example the Battle of
Magenta or
Solferino end as a failure, perhaps even with the death of King Vittorio Emmanuel II and Napoleon III deciding to just give up his Carbonari dream.
The Problem is that the most cohesive Liberal Republican Group, Mazzini's
Young Italy was discredited after their failure to take over Milan in 1853, not to mention that Republicanism was also somewhat discredited by the failure of the Roman Republic of 1849. But let's say that the Sardinian King is dead, the dream of a Sardinian-led Unification are over and Napoleon decides to move on to Indochina and America.
Fast forward to Austria clinging to Lombardia and perhaps taking some border provinces from Sardinia.
Fast forward to 1860 and Garibaldi, still bitter over the defeat at Solferino or Magenta leads an ITTL version of the Expedition of the Thousand, taking over Sicily with a volunteer force funded by the French and the British, and then marching on Naples as IOTL, as the Kingdom of the Two Sicilies collapses and with it the Bourbon Monarchy. Now Garibaldi is President of the Italian Republic and his next target is Rome.
Killing Cavour, the Italian Bismarck, might also help as it leaves Sardinia without a Prime Minister capable of keeping Garibaldi's Republic from invading the Papal States and the Habsburg Duchies in Central Italy.
Due to Napoleon III's relation with the Pope and the Republic's possible anti-clericalism, not to mention Austrian and Sardinian opposition, it's hard to say how successful this Republic could be, but maybe if the British support it to counter Louis Napoleon's ambitions on the peninsula, or if Napoleon III is preoccupied elsewhere and can't jump to Rome's defense in time, then it's possible that after conquering central Italy Garibaldi can overrun the Habsburg Principalities, which of course brings the problem of him having to fight against the Austrian Army without French support.