I tried the unofficial shop route years ago and wouldn't do it again - that cheaper battery ended up wearing out really badly. The truth is that in Milan there are very few authorized places beyond the Apple Store, and independent shops in the historic center basically don't have access to certified Apple parts, just compatible replacements which isn't the same thing.
What nobody tells you is that you can try MediaWorld or Euronics in the city - sometimes they have some certified Apple accessories, but for batteries and replacement screens unfortunately it depends on the store and what they have in stock. If the official price is really weighing on you, the only decent option left is to wait for some promotion on apple.com, which at least guarantees you're getting the genuine part. It's different if you like taking things apart and are willing to use verified compatible parts, but if you want peace of mind and a warranty then the official channel is what you're stuck with.