Your family is important to you, yes, but do you think they're treating you fairly? No. Do what makes you happy- and if it means having a relationship with this girl, do that. They'll either come around, or they'll use being bi as an excuse to judge you anyways- with or without the girl.
I say this with a wealth of experience trying to please my family: at the end of the day, the only person you answer to, is the mirror.
"She thinks... she can make people do what she wants or needs, what is right, by the sheer force of her own talent, not by forcing them... she can teach them and persuade them... that they'll catch it from her. This is still faith in their rationality, in the omnipotence of reason. The mistake? Reason is not automatic. Those who deny it cannot be conquered by it. Do not count on them."