I have my Philippine passport that will expire this July (2021) and I'm still waiting for my Permesso di Soggiorno to be released anytime sooner.
Please remember, Carta D’identità is (more or less) a symbolic ID card and you can’t travel with it or use it as an ID outside Italy (this should also be stated on the back of the card).
Furthermore, according to the rules, as a non-EU you always need to carry a valid passport. Even though, documents are not frequently checked between Schengen flights, you should always have it on you or if not you must have a valid justification or replacement.
One solution, as suggested by @Willeke is to contact the embassy and request for an emergency paperwork.