Italy can at times issue extended stay national visas in Italy for those who arrived without a visa, but I don’t know how convenient that is there. I just know that it’s been done. For at least some other Schengen countries, it can be way easier and faster to go back home to have it done and then return. Keep in mind that the airlines and/or Schengen passport control may want to see proof of departure from the Schengen zone within 90 days of arrival unless having a visa, a visa-substitute, a visa-waiver or residency permit letter/sticker/card.

What I would do is get the replacement passport, and then deal with Italian consular officials in the US. And if while doing this the flight is scheduled to leave, show up at the airport and inform the airline of the situation in person. Why? Because the airline reps at airports may be more willing to ignore/waive change fees.