We live in Putney and chose Thomas's Battersea for our children after looking round several co-ed schools, both those finishing at 11+ like Hurlingham, Prospect Hse, Thomas's Fulham and those finishing at 13+ like Thomas's Clapham, Fulham Prep and Thomas's Battersea.
For us Thomas's Battersea won hands down because it's a creative, busy place with a real ethos of kindness. We found, and still find at matches, Thomas's Clapham a more competitive place, and it seems to suit plenty of children/parents well, but we felt our children would thrive at Battersea and they have.
I'm surprised that you can't find much information on the website - it's much fuller than it used to be and has a list of Leavers' Destinations by year if that's one of your main criteria. I would urge you to settle on a school which has a wide range of final destinations. When we looked for our first, had you asked us, we'd have said co-ed day was what we were hoping for; he's just received two offers from top boys' boarding schools as that's what he's after ... he's changed a lot!
There's nothing like viewing the schools to get a feel. We knew we wanted the children to go to Thomas's Battersea when we got as far as the entrance hall; we walked in, no child held the door etc which I would have considered bad training except the children whizzing in every direction were so excited by what they were doing and beaming as they sped between lessons that the atmosphere just won me straight over. You could just feel the good cheer and warmth of the place. (Yes, a Yr 6 guide did hold the doors, show us round most charmingly and enthuse about every room/lesson/teacher in the school after that!)
If you're after reasonably close 13+ prep for girls there's not many to look at so set up some appointments for September .... 