I have found that I get teased a lot, but I can point out that it isn't fair to do so, and it ends quickly. She may not be the right fir if she can't accept all of you. I don't know about your relationship, but maybe you can talk to her about it.
Despite my ex being bi-curious himself, he would occasionally comment that my bisexuality made him worry that I would cheat on him with a girl. He didn't trust me as much as he would have trusted a straight partner. My current boyfriend on the other hand is more than happy to have a bisexual girlfriend. It makes threesomes a little easier to arrange!
I agree that you may want to take a close look at your relationship if your lady isn't ok with you being who you are.
My husband is absolutely fine with who I am. He knows I won't stray and that if we ever engaged in a threesome it wouldn't mean anything more than just sex.