The problem with this alternate Vienna Congress is that none of the Great Powers (other than Austria) has a real interest in Italy: therefore it makes sense to hand Italy over to Austria as its own private garden, and keep hands free in Germany (where all of the GPs have real interests).
Religion has little or nothing to do with the outcome, and even the principle of "restoration" works if (and only if) there are no good reasons to do otherwise.
OTOH Austria has not a strong interest in re-drawing completely the map of Italy: Veneto is certainly important to them (both to have an unassailable link to Lombardy and because it would be a province which is a net tax-payer), and it's reasonable to reinstate Habsburg proxies in Modena and Florence. There is no need or desire however to partition the Papal States: why should they? Post-Vienna Austria is the natural protector of the Papal States, while partitioning them would potentially create problems: an enlarged Tuscany might be less obedient to Austria dictates, and creating a principality out of the Papal Legations in Emilia, Romagna and Marche introduces a wild card (who would be the prince chosen for this new state?).
Therefore restoring the Papal States is the default solution: it is not controversial, follows the principle of restoration, keeps the Catholics happy and does not allow other powers to ask for compensations elsewhere. Metternich was pretty sure it was an optimal solution, although he had to change his mind 20 years later when he was forced to recognize that the absolute incapacity of the Church to grant a minimally decent form of government (and the acceptable bar was pretty low) became even too obvious.