Frankly speaking, given that you don't have much experience, don't have much time, and you haven't been in Italy for so much time, I would skip diving.
Italy usually doesn't offer the same scenario of the Maldives or the red sea, except for some places (for instance, Ustica), and this can be disappointing for many divers. There are many exciting things, but they often require technical training, for example, the Haven wreck or the caves in Sardinia. But given your level of training, many of these amazing things are out of your possibilities.
On the other hand, in the dry part of the world, Italy has to offer so much; close to Naples, there are so many unique places to visit: the Costiera Amalfitana, Pompei, Naples itself, Capri, Ischia and the amazing Reggia di Caserta (imagine a small version of Versailles).
If you really want to dive, I would think about the experience in the submerged city near Naples, but it's up to you to decide whether the cost of bringing the equipment is worth a single dive. You may try to get in touch with some local diving centres if you want more details, but obviously, they will try to sell their products, no matter what...