Obviously it's good to study at a good business school, though it's naturally not as strong of an ability signal as studying at a good economics undergrad school, when applying for postgraduate economics.
Yes, the GDE is open to people from most degree backgrounds, so you should definitely qualify and meet the entry requirements. It might even be the case that you find the start of the GDE easier as you have a business school background whereas some will be coming from a complete range of undergrad degrees so at least you have some intro classes.
Regarding the student experience, was that anecdote from someone studying economics? Ultimately master's degrees are more about building your research skills and a lot of that is solo work. I've heard LSE has the worst student experience as it's all bunched into a tighter time frame (most MSc's have classes from Oct-Jan, Feb-May/June, then your dissertation over the summer and hand it in end of aug/start of sept. At LSE you have to do your dissertation alongside the 2nd semester, though it's shorter at only 6k words).
While I went to LSE, I think generally Oxford and Cambridge have the best student experiences. A lot of people don't want to study in London as they're working cities rather than student cities, and are very expensive, often students can live quite far apart. Many complain that Warwick is in the middle of nowhere, the nearest largest places are Coventry and Lemmington Spa, which aren't exactly the most desirable places for student life, though if you're a fan of the countryside then it could be a good option.
Oxbridge often have the best student life as they're beautiful cities and you're surrounded by history, tradition and amazing architecture. They also use the college system which makes building social ties a lot easier. They also have a lot more prestige than the others, many say that LSE is better at postgrad level (incl me) but Oxbridge is definitely more prestigious, though I doubt it would make a difference when it comes to applying for jobs.