Perhaps if the area around Rome is left to the Pope, the conquest of the other territories would be acceptable?
Not really. Wars involving the Papal States unanimously had the Papal States as an aggressor, and when the Pope got involved it was always as part of an alliance to keep Italy balanced (to prevent one country becoming dominant). Popes never tried conquering land, they always claimed to be fighting simply to keep Italy free of domination. This isn't to say that they would always win, of course, but the Papal States really were off-limits for capture. As a combatant Rome was at times open to attack, but this was only spoils of war. Actually trying to take land off the Pope would be universally condemned and would probably prompt France or Spain jumping in to revert the land swap. The first time the Pope properly lost land was the last time it happened - when Italy unified, at a time when people no longer believed in the Pope's right to temporal (non-religious) authority. You won't get that attitude before about 1800 though. For this reason, while the Pope is in Avignon the same stands - then again, most of the time the Pope was in Avignon there were two Popes. Any successful conquest of the Papal States in the name of the Avignon Pope would oblige you to cede the entire Papal States to the Avignon Pope as "his right of office" so no potential for landgrab there.
You do have an option in about the 1300s, though. The Pope's authority had slipped, leading to a lot of people, such as Papal condottieros (mercenary military commanders) exerting their authority claiming to be Papal representatives. The Pope kept control over roughly what is nowadays known as Lazio, but the rest of the Papal States broke up into what were essentially nominal Papal vassals as the Pope was in a weak position. If these usurpers were ever threatened they could always claim to be under the Pope's command, which was an argument which essentially could not be contested - even if these lands were conquered, the Pope would demand them back anyway. Since these men never technically claimed independence from the Pope they survived for about 50 years before the Popes had strengthened their position and were able to re-exert control. Say the Pope invades to re-exert control and fails badly. If you can have an uneasy stand-off between the Pope and his supposed vassals then there's a small chance that eventually the Pope can publicly relinquish suzerainty over the land in the hope that someone else (Milan
par exemple) will do the hard work for them and give the Papal States the land back as tribute. That could give you the chance to let the entire Papal Adriatic coast go free, which also lets you link North and South Italy up.
But it's a really dicey thing, mainly due to the way the Pope is never supposed to be challenged. Personally I would just limit early Italian Unification to North Italy.