Are you someone who doesn't like to be friends with a fan of the same group member?
Are you someone who thinks of a fan of the same member as a rival?
私が知っている限り、「同担拒否」に相当する英語がないと思います。ですから、そのコンセプトを英語で説明する必要があります。
多分、「someone who doesn't like to be friends with a fan of the same group member」という説明が一番近いと思います。
「あなたは同担拒否ですか」= Are you someone who doesn't like to be friends with a fan of the same group member? / Are you someone who thinks of a fan of the same member as a rival?
Do you dislike fans whose favourite member is the same as yours?
Do you hate people who support the same member as you?
残念ながら、英語では「同担拒否」に当たる、コンパクトな表現が存在しないから、直訳できないですけど、伝えたいことを色んな風に言えます。
例えば:
'Do you dislike fans whose favourite member is the same as yours?'
'Do you hate people who support the same member as you?'
'Are you someone who refuses to become friends with fans of the same idol?'
上記の例文は、文の構成や単語がちょっと違っても、全部「同担拒否」を説明できる質問です。憎さの程度の違いがあります: 'to dislike' は「好きじゃない」ということで、 'to hate' とか 'to despise' は「大嫌い」という、強い感情を表します。'To refuse to become friends' は「友達になることを拒む」という意味です。