The first thing to know is that Alex's former house wasn't a mansion. Although, that imaginary news article said it was a mansion, it was a smallish house. It was a nicely designed 3 bedroom, but well under 2000 sq ft. If that's a mansion, then I used to own a mansion in my modest middle class suberb.
Alex was house hunting, but it was for a rental, no doubt to use during filming for BLL. The lousy paps gave away the home's address, including a shot of his jazzy BMW in the garage. A simple google search indicates the home was available for leasing in Bel Air. It's sales price is under $3M since it is also modestly sized.
BTW it's actually hard to picture a $4.5M home in Silver Lake, which is slightly down market. That actually would be a mansion. It's close to Los Feliz (where his former house was), and both have hipster appeal because they are older, cheaper neighborhoods.
And no reliable news source has picked up that house purchase story, which includes quotes that sound like fabrications. Alexa spends quite a bit of time in London. A house in LA seems considerably inconvenient.