The ATR-72 can be equipped for ASW (I think it was designed in, but not chosen by Italy: I'm not sure if any other ATR-72 MPA operators have fitted it), & doing that could be the quickest, lowest-risk way to upgrade Italian airborne ASW capability. But that'd still be rather limited, & realistically only an interim upgrade pending the introduction of more capable aircraft.Maybe an ASW version of the HC-27J and MC-27J is the fastest and safest choice. Besides that it can be developed domestically.
I expect that the P-8 could be got into service earlier than any other full capacity ASW aircraft.