What style does your group practice? When you say club, are you a formal dojo or a group that meets up? There's a lot of ways to describe these groups and there's some nuance to the words you choose. For example, "~kai" may have a more formal "organization" sound to it. If your club is tied to a specific location, then dojo seems more appropriate in the name.
My first thought was if you were already married to the name "Somerset Ryu" and a "non-denominational" group, why not use サマセット流, as that "ryū" is the one used to denote a karate style, and it's a homonym for dragon. So, this would be the "Somerset-way/style" karate club. I'd then make a club logo that uses the red dragon from your county flag, making it a good pun to boot. This ryū does make it look a bit like a formalized system, but that confusion would only be from people reading the name on its own out of context; you wouldn't say that you practice Somerset-ryu the same as Shorin-ryu or Goju-ryu.
If the name is more flexible, you can tell us more about your club to give us a better chance to find a good one. I like
@Toritoribe's suggestions, and the term 倶楽部/kurabu is the kanji transliteration of "club," which sounds like a somewhat serious regular meetup, maybe at the local community or recreation center. There's also circle/サークル which is like a club but less strict and formal. I was in a Wado circle in college, so i'm not sure if there's a "school" connotation to it or not. Bukatsu (short for kurabu-katsu, club life) is huge in school, comparable to fraternities and sororities in the amount of time and dedication they demand of their members. Circles are much more relaxed and easy-going.
Also, karate-ka in
Okinawa seem to like the term "kenkyūkai" which literally means study or research group. They strongly see their practice as a study, research, investigation into the art, their own body, and self defense.
Lastly, if there isn't one already, you could just go with something like サマセット空手倶楽部, which is just "Somerset Karate Club," and save the dragons for the logo.