The Pope who got closest to "uniting" Italy was Alexander VI, mostly thanks to his son Cesare conquering everything he could get his mitts on, though that was all way back in 1492-1503.
Hrmm. For an early 19th century POD, maybe you could have Cardinal Consalvi manage to have the Papal States be awarded some more territory at the Congress of Vienna. Maybe some chunks of Tuscany (the State of Presidi, maybe) get rewarded to the Papacy and they use it as a springboard to compete with the Savoyards for dominance in Italy, probably by embracing (rather than largely ignoring) the concept of Italian nationalism.
Only viable way i see is Victor Emmanuel giving the pope a place that is similar to today's English Queen to gain more legitimacy but i dont know how plausible is that.
I'm not entirely sure, but wasn't one of the early plans for a unified Italy an extremely loose confederation of states under the Pope? somehow get rid of the Savoyards and you might be able to get all the conservative monarchist nationalists to fall behind it.
True but in the history books it would be stated that the first head of state of the modern unified Italy was the Pope therefore it would satisfy the question in hand. Also i believe that the Pope would not accept being a simple figurehead but would consistently get involved in the everyday politics by influencing the majority of Italians that remained and mostly remain faithful Catholics. We see something similar in today's Greece especially with the last Greek Archbishop who initiated protests and affected election results. Considering all those facts not only a unified Italy under the theoretical leadership of the Catholic Church would not be viable but would diminish what is left of the influence of the Pope.The "reduction" of the Pope to an absolute figurehead within the hierarchical structure of a unified monarchical Italy would benefit the Papacy greatly. A socio-religious co-op of the Vatican would grant the Italian king more legitimacy and stability. Such a move would likely avoid the defiant papal "imprisonment in the Lateran". This move would also reassure the Papacy that its Roman holdings beyond the Vatican hill would remain firmly in the grasp of the Church. Any symbolic union between the papacy and Italy would likely dissolve with the establishment of a Republic as in OTL.
Any republican government of Italy (as in OTL) would render the Pope a titular monarch utterly devoid of even feeble figurehead temporal influence. The dissolution of the Papal figurehead after a dissolution of a monarchical Italy might jeopardize the papacy's Roman holdings, however. Still, this ATL system of a symbolically temporal Pope would not and could not last forever especially with the conversion of many European states to republican forms of government after the world wars. Remember that Paul VI dramatically rejected the temporal power of the Papacy by shunning the tiara soon after his coronation. He ushered in the modern and sole role of the Pope as Bishop of Rome and shepherd of the Universal Church. The modern popes' sole use of the mitre as symbol of pure spiritual leadership was an inevitable development OTL and probably in this ATL as well.