I don't like Milan enough for 3 days. Have you been to Venice?
I guess early September is OK, but I'd make sure your top targets in the Lake district are open for your visit. (I haven't been there.) The date makes sense in view of how hard it is to book in highest season.
Northwest Belgium is discussed a lot here. You can get A LOT of information from the good search engine, top center. I'll just note that unreserved train travel is so easy that you don't really HAVE to change hotels to see two cities. Many people consider Bruges a "must sleep", but I personally find it okay to daytrip to from Antwerp. Note that Paris is well connected to Antwerp, with non-changeable, non-cancellable advance-discount tickets that save a lot of money. Edit: Brussels can also take you to Paris, but the choice of trains and departure stations can be completely different. May work for you, maybe not. But I prefer Antwerp to Brussels, except for those who need to see the Royal Museums of Fine Arts in Brussels. I have daytripped to Brussels from Antwerp several times.
This is only one of many posts, but it starts to discuss your issues and is very recent:
https://community.ricksteves.com/travel-forum/belgium/bruges-or-brussels-as-our-base
If you have the time and energy, it is very hard to exhaust Antwerp's attractions. For example, there's the first-rate outdoor sculpture garden that's a bit hard to reach on foot, the Sunday-only visit to Lillo, an abandoned village, families picnic-ing by the river on weekends, the foot tunnel under the Schelde, the roof of the MAS museum, Reubens' tomb, Vrijdag market, all kinds of things that many visitors never get to.
You have enough time for three locations, but I question how far it is from Milan to Antwerp. You should look into the air connections before you plan too much more. Note that Ryanair probably doesn't fly to BRU, but to an inconvenient secondary airport for Brussels. If Venice has better connections, I'd choose that in a minute. Would you consider someplace closer, like Cologne or even coastal Germany? I can't recommend Ostend, Belgium as a substitute for the Italian lake district, it's not like that at all.