Here are the people who can check your passport on the way:
- Airline check-in agents. Will check your paperwork against Timatic, the database used by airlines for this purpose. See below.
- Origin country passport control - none in the US, and even when there's one, they usually don't care about other countries' requirements.
- Airline gate agents, if the check-in agents did not check, same rules as above.
- Transit country passport control - does not apply if you stay "airside", which is apparently your case (double check you don't have to switch terminals, though).
- Destination country passport control - at this point obviously any requirements for the transit country are not their problem.
In your case (I'm considering you have US passports, let us know if I'm wrong), Timatic says:
Travelling to Israel via Italy
Travelling to Israel
Passports and other documents accepted for entry must be valid for the period of intended stay.
(no requirements regarding passport)
So the only requirement is that the passport is valid for the the period of intended stay in Israel. No requirement for Italy.
It is possible that the check-in or gate agent may initially be confused (especially if they are new) as they will be applying the Italy rule for many (most?) people they're checking, but they should quickly revert to reality once they realise you are in transit. If they don't snap out of it, tell them to "check Timatic".
Note however that there are quite a few Covid-related requirements for both Israel and transit through Italy I have removed from the quote, as this wasn't relevant to your question. You may want to review those.