Well, i would say it's likely that soldiers prayed God to succeed in battle given that basically all of them were Christians. That being said, every man of the Renaissance knew the Papacy was of double nature: an Italian princedom in temporal matters and the universal leadership of the Church in spiritual matters. So, to the Italians of the time, it wasn't something new nor much of a scandal that wars could be waged by or against the Pope.

Sometimes, however, when Popes went too far, a different sentiment may arise: rulers at war with the Pope may complain about the temporal power of the Church (whereas when they were loyal to the Pope, they would exalt it) and/or regard the particular Pope of the time to be unfit (this until they signed a peace treaty; anti-Papal propaganda usually ended with the war). This sentiment is more commonly found in foreign monarchies at war with the Pope rather than in Italy, where people were more used to it and accepted it. Italian or non-Italian catholic states at war with the Pope would still recognize Papal authority in moral and spiritual matters. (note that there were exceptions: some Renaissance monarchs tried to depose popes, like Louis XII of France).

A bishop of a state at war with the Pope may support the country in which he is (again, remaining loyal to the Pope in spiritual matters) or stay neutral or even support the Pope, depending on the situation. This may depend on wheter the Bishop was an actual resident in the city in which he was assigned (not so common prior to the Council of Trent) or wheter he was primarily in Rome. I'm not aware of a Bishop blessing an army going against the Papal State. But this brings us to another topic:the inderdict. If a Pope launched an interdict, like he did against Venice, important religious rites could NOT be performed by the local church. And if a Bishop broke the interdict, you can be sure the Pope would do work to replace/depose him. This was a very powerful tool in Papal hands.

Also, we have to understand the nature and scale of warfare during the Renaissance. A country like France, the most populated in W.Europe at the time, deployed about 10.000-20.000 men in very important battles. Who fought battles? The nobles of the sword and their entourage (the "national" part of the army) + mercenaries from all over Europe (even in French armies, there were Germans, Italians, Swiss etc). Now, being states of small size, the Italian states lacked a proper "national" army. They relied on Italian and non-Italian mercenaries usually led by condottieri, Italian military leaders contracted to fight wars. So they fought for money and greatness; they were very religious but religious feelings did not necessarily dominate their actions.

Sources: ''Roma, Romae, una capitale in età moderna'' by Marina Formica. "Il Sovrano Pontefice" by Paolo Prodi.