@collegemom3717 The “backdoor” assertion is nonsense. It is utterly irrelevant and will not factor in any decision.
In the case of a student applying to transfer from Barnard to Columbia, you will need compelling academic reasons since you already have access to the resources of both Barnard and Columbia College, can pursue a Columbia or Barnard major, and your diploma will come from Columbia University… The big difference is Core and so you could say you want to transfer to take Columbia’s Core. You’d need to show how essential this is.
Arguing about Columbia having better financial aid also isn’t a good idea, since Columbia does not guarantee financial aid to transfer students and, moreover, the only major difference is that Columbia doesn’t include loans for low income students. So that is unlikely to impress the admissions committee. You could wind up with less aid.
One other thing. You MUST have the support of your Barnard adviser and should speak to a Barnard Dean. This is clearly stated in the transfer admission policy. Columbia will contact Barnard, so if you don’t do so, you will have a problem. And of course Columbia will know that about your previous application and will know why they didn’t offer you admission to begin with.