I knew I would be relevant one day
I've just been to Essevierre (in June) although I flew into Verona so my experience is based on that.
I went over with nothing more than a SG certificate and an EFP. Italian police didn't even bother looking at the latter and simply checked the gun number against the SGC. They asked why I was bringing a gun into Italy to which I responded to shoot at Lonato and visit Essevierre to have a stock modified (at which point I showed them a piece of paper with the Essevierre address details), to which they just shrugged and said OK. If push comes to shove just ask them to call Essevierre who should be able to confirm you have an appointment with them. They should be well versed on shotguns coming into the airport as the area is gun (shooting and manufacturing) central!!
When you arrive you will need to ask in baggage reclaim where you collect your firearm. At Verona you have to leave the arrivals terminal and go to the police/security office in Departures to collect your gun.
Not sure what its like at Milan but at Verona you are required to pay the airport a 40 euro fee to transport your gun from the check in desk to security and then on to the aircraft itself.
All very simple although it can take some time. Make sure you are at the airport in plenty of time on the way back.
Harry and I had a ball when we went over so hopefully you will as well.
PM me if you need any more info.
DT
Edit: Oops, it seems you already have while I was typing this.