You can try adding your Italian QR code to Stocard (on PlayStore: https://play.google.com/store/apps/details?id=de.stocard.stocard) search for "Certificazione Verde COVID-19" as a "Card" and add it there. See if it shows it as a booster or simply as a first jab. I am curious, mind trying and reporting back please? If not, a decent pharmacy should be able to accept your third as a third if you show them all and add the third only as a booster and provide you with the correct QR.
You could download Stocard on your phone (for Android: https://play.google.com/store/apps/details?id=de.stocard.stocard for iphoe look it up) then as countries, select Italy to see "Cards" available there. Then search for one "Card" called "Certificazione Verde Covid" scan the QR code you received on paper with it and you have a fully fledged and working pass that you can use. See if it shows as a full vaccination and if you can get in shops with it (mine works perfectly in shops). You can do this, before "risking" adding it to CoronaWarn app. I had my booster but not yet since 2 weeks so I am not adding it yet to the coronaWarn either and I am using stocard instead and it work perfectly. hope it helps