To board your flight to Japan, you must show your airline that you have enough documentation to complete your trip. Airlines use databases to verify what documentation is required. If you complete the form here with your information, you will see that a transit visa is not required. if you stay in the transit zone. However, as you say, you will have to leave the transit area to register for your second tranche. The form above gives this guide:
A transit visa is required if the passenger leaves the airport transit area.
Valid visas will be accepted in the expired passports provided they are officially linked to a valid second passport.
Passengers traveling through Japan can obtain an entry permit at the discretion of immigration personnel. They can not leave the area described by the staff member, and other restrictions may apply. Maximum stay of 72 hours.
So, in fact, you will need a transit visa. The third paragraph offers an alternative, but Alitalia may not allow you to address it anyway, as they may be fined if they deliver it to Japan with insufficient documentation.
In conclusion, if you need to leave the transit area, you must have a transit visa.