USA money exchanged at banks?

A lot of banks these days only change currency to account holders. But there are lots of exchange bureaus like Travelex, which can be found in big cities, train stations and airports. But of course they have charge a commission, so no matter what you do you, you loose a bit. ATM will always give you a better rate, even with a $7 fee use you debit card to cash from the ATMs.

Ask your bank to temporarily increase the daily withdrawl limit. That way you can withdraw more money fewer times, limiting the number of times you are charged the $7 fee. But sometimes the ATM has withdrawl limits too. Exchanging money at banks typically costs about 10% over the interbank exchange rate and not all banks do it. Using a currency exchange office typically costs 20% or more. Even if you only withdraw $200 worth each time, at $7 that is only 3.5%. The Italian bank ATM will not charge you a fee.

I'm assuming you don't have enough time to explore setting up a new account with an institution that won't charge you so much. That's too bad, as there are several out there like that. Read Rick's Travel Tip on Cash and Currency

If you have two weeks before departure, that may be enough time to open an account with another bank and get a debit card. If not, I would follow the advice to increase daily withdrawal limit and take the max out each time.
That would still likely be cheaper than bringing US dollars to exchange there.