Have you and your D visited Barnard? Is she highly attracted to Columbia's Core, and not Barnard's Nine Ways? Because she would not be able to take the Core.
Barnard is NOT a "back door" to Columbia and I hate that anyone told you that. If your D truly will be always trying to make herself out to be a COLUMBIA student (implying Columbia College, not the general Columbia University student body, of which she most certainly WILL be a part as a Barnard student), then she probably needs to take advantage of one of her other excellent opportunities.
And yes, only putting "Columbia University" on a resume would be deceptive. "Barnard College at Columbia University" is, however, entirely accurate. If she actually attended Barnard, though, I would be willing to bet that she would quickly decide on her own that she is happy and proud to attend Barnard.
My D was initially attracted to Columbia. But when she visited Barnard, interviewed and took the tour, she quickly changed her mind! Has not regretted that a day since!