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Explanation: Really depends on what kind of "club" you mean by "Dragon Club". There is a popular music band "Dragon Ashe", which reads "Doragon Asshu" .
Japanese people usually know what dragons are and they say "Doragon" (as in Dragon Quest (Doragon Kuesuto) and other games and animation characters). If this is for a club, whatever the club is for, it sounds more hip to say "Doragon Kurabu" , than to say "Ryu no Kurabu"; But!!!
If you don't care if it sounds hip to Japanese ears, you might go with "TATSU NO KAI" or, to make it sound a bit delinquent or gang-like, you might translate it, "TATSU JUKU", "TATSU GUMI", "RYU JUKU", or "RYU GUMI".
Explanation: I agree with the native Japanese above that you would be better leaving "Dragons" as its phonetic equivalent but, like the famous Japanese Rotary Club equivalent, "Lion's Club," (ライオンズ クラブ）, I think that you would be better keeping the 's which is commonly pronounce "zu."