This is a tough one for sure. I teach English here in South Korea and my students often say that they "play" 태권도. I always correct them because it is definitely wrong to use the verb "play" with this martial art. They do it or they practice it. As for the noun though... Player is absolutely wrong, in my opinion. I haven't found the right noun yet though. I will be following this discussion with great interest. :-)
I know there's another word for athlete 체육 선수, and for competitor 경기사/ 경쟁자.
Players play games. Athletes do athletics. Competitors compete in competitions. And, of course these things overlap.
Anyhow, we usually just refer to them as club blackbelts. For example, "Bob is a third dan blackbelt from Kim's Taekwondo." Or, "I am a blackbelt from Yongin Taekwondo."
Or it would be even easier to drop the be-verb. I do Taekwondo. Ha ha.
In Japanese you'd never say that; you'd say bujutsuka->moosoolga. See the exact-phrase hits on Google: "무술가" ~261,000 hits "무술인" ~118,000 hits
As for player, you wouldn't use that with boxers, wrestlers, basically any fighting sport in English. "Play" is used for sports when they're games not purporting to any realism. Maybe then it's somewhat appropriate though, as many people are just doing it for fun and not so serious about it. 선수 though (選手=senshu, literally something like "selected for skill") is only used in Japanese for professional athletes so that must just be silly . . .