Upvote:3
As long as you're not going through border checks (i.e.: leaving the airside area of the airport), you should be fine. Since you're traveling from outside Schengen to outside Schengen - you shouldn't be going the border checks at any point. If you want to spend some time in Rome while waiting for the connection, then you may have an issue, but with 4 hours connection it's probably too short.
Flights to Israel go through additional security, so be at the gate early.
Upvote:3
Here are the people who can check your passport on the way:
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
Passport
Passport required.
(...)
Document Validity:
Passports and other documents accepted for entry must be valid for the period of intended stay.
(...)
TRANSIT INFORMATION
Transit Countries
Italy
Passport
(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.