I actually see this as hope that she will come to accept your trans-ness in time. Try to see it their way: Your given name is special to your parents, because they were the ones who gave it to you. Your cousin is much less close to them than you are, so changing how they think of her is easier for them. They may even find the idea of having two trans girls in the family all too unlikely, and think that you're just inspired by Brooke and not trans in your own right. And, most of all, your cousin has (I assume) been out as trans for longer than you have, so everyone has had more time to adjust.

So yes, for the present it seems like your situation is a bad one. But what I see here is hope - if your mother accepts your cousin as Brooke now, she certainly will accept you as Ayumi in the future. Just give it time, and keep working for it.