How much time since you posted it?

I don't know the procedures in Italy, however in other countries you would certainly pay the VAT inclusive price upfront, then get the VAT refunded to your card, but this can take weeks or months to be processed when you send the form back.

Since they said they wouldn't charge you the VAT at all upfront (this surprises me, but I don't know the workings of the Italian system), then raise the matter with them (and do this now).

If they come back saying that when the form is processed, this will be refunded etc, then I'd accept that, and that it will take time.