AWESOME TRIP
Hola!
though requirements. I think you have to yield on some areas.
For flight it's easy to say Vueling is your best choice.
/// Sep 18-23 165€/330€ ///
/// Oct 3-8 135€/270€ ///
Direct flight and reasonable traveling times, can't really expect anything better.
This however leaves you max (600€-330€)/5 nights = 66€/night. Will you find a 3star hotel from Center of Barcelona with 66€/night? No. For a 'proper' hotel there's (only) Travelodge Barcelona Poblenou (260€) - just one star but still look's decent enough. For ***+ standard you'd need 450-500€ (some excellent choices e.g. Derby/Derby suites, SM Hotels and Astoria).
So nothing under 350€? There's plenty of awesome guest houses (and apartments) in Barcelona to choose from. Check e.g. Microtel Bauhaus (300€, right in central). Go with one of those and use money on wines and tapas, you won't be disappointed. Spain is known for excellent quality hostels/guest-houses.
Hope this helps. Have a good trip!