I think how you act around her is how you act around anyone that you don't exactly get along with but is tied to someone that you want to maintain a relationship with...you treat her kindly and respectfully. You don't have to be her best friend, but continue to be pleasant.
I don't think there's anything wrong with saying she's from a country you and your brother lived in as a way to facilitate conversation between them (I'm assuming here that the when she told you she was from there, that you also connected that you had once lived there...how did she act then?).
I don't think it was polite for your brother to accuse you of lying to him...especially in front of her. A better response would have been "Oh, I thought my sister said you were from there. I must be mistaken." And a better response for you would have been "Oh, I thought you said previously you were from there. I must be mistaken." I agree it was rude for you to talk about her as if she weren't there in front of her. And I think insulting to say "She IS from there" when she just said she wasn't. (Regardless of what she said beforehand).
And her acting like nothing had happened the next time she say you, was the correct response on her part.
Not everything and everyone can be explained. Who knows why she said she was from there once and then denied it the second time. There may be a legitimate reason and there may not be. I don't think it matters if she lied, when she lied, how stangely or not she acted...the response from you needs to be the same. Don't argue with her, don't talk about her as if she isn't there, and be nice. It's all you can do.