There are always a few students who successfully transfer from Barnard to Columbia. It’s not easy, of course – your application has to be strong. But if you can put together a strong application, then why not?
As far as the whole Barnard-Columbia issue people are referring to, I don’t believe that Columbia cares about that. I don’t believe the Columbia AdCom will view your application with extra favor nor with extra scrutiny because you are applying out of Barnard. It would be no different than if you applied out of Smith or Wellesley. Assuming the rest of your application is strong, then your motivation is actually stronger than students from those schools since you are better able to expound on why you wish to transfer.
It would be a “tough transfer” as someone stated. But it would be tough because transfer is tough in general, not because you are applying from Barnard.