Can the Sports Council, Local Authority or Sport England or anyone else for that matter, insist on membership?
No. This would contravene the law. You should be able to choose which organisation if any, you belong to. However if you wish to play Sport Karate, then you may have to join a Sport England/Sports Council recognised National Governing Body to be permitted to compete. If you don’t wish to compete, then you don’t have to join a NGB. TKGB is only concerned with Traditional Karate and not Sport Karate so this has no relevance. You are free to choose with whom to affiliate.