City breaks in extreme heat are not for me. But with only 3 nights, I would not want to waste time getting somewhere far from an airport.
Venice. Not in July. Crowded like Disneyworld during term break. Hot and muggy.
Palermo - amazing place - will be shocking hot in July. Beaches are close by, food is incredible. If you can take the heat, this is a great choice - close to an airport, very beautiful, undeniably an 'Italian' experience. Easy to organise a day at the beach (you will need to reserve sunbeds at a stabilimento in advance!). Ditto Ortigia (Siracusa). Touristy in the best way, on the water, plenty to keep you busy for 3 nights, romantic. Depends how you get there (flying into Catania then transfer?).
Your partner wants to go to Naples. I love it - but again I would be tempted to stay at the beach and travel into Naples for an afternoon and a night out. Avoid Pompeii, because... hot and crowded. Herculaneum instead if ruins are your thing. I think you could have a fabulous 3 nights there. It is more chaotic and keep-your-wits-about-you than other cities. But the food and the energy make up for it.
Puglia. Any excuse to go to Puglia is a good excuse. Great beaches. Amazing hill towns (organise a tour). Even better wine.
Rome - not a bad choice. Yes, crowded, also hot, but it's a big city and can take it - better on this score than Florence. There are good beaches nearby. Very hot, but that's a theme here, eh?
Florence... not in July.
The lakes in the north - great idea. But you need to get to them. And I worry about that taking up your time. Depends on how you organise transport from Milan.
Same with Cinque Terra. You need to travel on from Genoa to get there. Worth it if you don't mind that.
Hope you and dp have a good trip!