If the CP had won WW1, it is quite likely that A-H would have resurrected the Papal States (although it is quite likely that they would have not been a republic).
After WW2 it is not possible: the USA need the church to back the Christian Democrats against the communists, and imposing the Papal States once again it would have been the best way to have the rest of Italy going red (I would expect that a resurrection of the Papal states would have been quite controversial and unpopular in the USA too).
The first Roman Republic (after the french revolution) might have lasted only if the French revolutionaires had remained "pure". No Girondins, no Directorate, Napoleon accepting to be just a military commander for the Republique (the last is clearly ASB
).
The second Roman republic had also no chances, unless the 1848 revolution had been truly successful. I would like to live in a TL where this had happened, I am sure it would be much better than OTL. OTOH, such a republic would have clearly enforced a strict separation between state and church: even if the pope had remained in Rome (very few chances of that), he would have been restricted from playing a political role.
Maybe a (much) better Pio IX, truly committed to liberal ideas and willing to step off from the throne of temporal power. Very very ASBish: even if the man had been willing to consider it, the church structure would not have accepted it. At best, it would have resulted in a catholic schism.