I'm not sure about in the Henbury/Pilning/Bradley Stoke areas I'm afraid. However I can definitely recommend the kickboxing classes at Clifton College (taught by a world class champion, def worth travelling for!) and also the boxing classes mentioned on this blog look good as well (http://www.cliftonhotels.com/news/new-year-new-you/) and perhaps marginally closer to the areas you mentioned as well!
Hope this helps. if not then good luck with your search!