It looks like you've been doing some really good research already, so because I don't know much about badminton at uni, all I have to add is that it's very common for nearby universities to have reciprocal agreements that allow students to join each other's societies.
So, if (for instance) UWE had a very good badminton soc with coaching, you could go to Bristol Uni but still take advantage of all that UWE's badminton soc has to offer, or vice versa. The only downside to that is that you can only play in BUCS leagues for you own university, so you wouldn't be able to compete for UWE at BUCS if you were studying at Bristol.
The Uni of London also normally has sports societies at collegiate level (UCL, KCL, LSE, QMUL etc.) but then an intercollegiate sports soc which takes the best players from the collegiate level sports socs (I don't know if this is the case for badminton though - you'd have to investigate).