1848-1849: Kindgdom of Sardinia lost the First war of Independence against Austrian Empire. King Carlo Alberto abdicates, and his son, Vittorio Emanuele II, is not interested to unificate Italy, that still divided.
What would happen?
Giving that Italian nationalism was really something at this point, you could expect someone more skilled and able to see how it would increase his own power as well going by a popular demand taking power.
Sardinia remains the most obvious candidate, but let's not overlook endless revolts in Italy that would make Eastern question look simple to manage for Austria.
Franz Joseph I, Emperor of Austria and King of Hungary and King of Lombardy–Venetia would most likely advise his son,, Archduke Ferdinand Maximilian or his younger brother, Archduke Karl Ludwig of Austria or his uncle Archduke Rainer Joseph of Austria (who is already viceroy of L-V) to take the position as sole King of Lombard-Venetia.
If Archduke Ferdinand Maximilian, is crowned King of L-V he is less likely to be killed like he was in Mexico.
FJ didn't trust Max. That was thanks to the British preferring Max to FJ, going so far as to suggest that perhaps the more liberal Max be given charge of Hungary after 1848. So I don't think he's gonna like the idea of Max being king in Italy - which is essentially still next door.
Maximilian was actually in secret negotiations with Italian nationalists to become king of a unified Italy. When FJ found out about these negotiations he removed Max from Italy.
So, both the Italians and Max seemed to be able to at least live with each other. The biggest resistance for such a deal would come from FJ, but it is not impossible with some creative meddling with the Habsburgs and the foreign political situation of the time.
With a Kingdom of Sardinia not interested in unifying Italy, in my opinion not only Austrian Empire can be interested in the enterprise, but only Papal State and Kingdom of Two Sicilies cand be strong peninsular powers.