There is no appeal process.
I didn’t want to point it out, because I tried to be kind, but it seems you need blunt assessment in addition to kindness, so here it is; your application was rejected, not deferred.
You think you’re a perfect fit for them and they don’t think so.
We don’t know why.
They probably had 3 times the number of qualified applicants. Perhaps they were looking for an oboe player and you play the piano, or you’re a ballerina and they want a gymnast. Your achievements and the Oboe player’s achievements are equally strong, but institutional needs (what Barnard needs) mean that they take the oboe player.
There may also be a reason for it, such as a problem in your essay or a recommendation, since your GC was rather confident in either an admission or a deferral (if your GC was flat-out confident in an admission, they overshot: with universities that selective, no one is a shoo-in). The best thing you can do now is making sure the same problem doesn’t crush all other applications (beside flagship etc.) and have someone review your essays (some of us here can do that if you want an external review).
Your GC having connections will NOT change a rejection. It might have done something for the RD round if you’d been deferred and she may have been able to talk to someone so that they review your application with an eye for something specific, but in this case, it’s done.
(Thinking that an appeal +GC might work could show two things: great cluelessness or great entitlement; what you’re talking about is telling Barnard professionals that YOU know better than them who they want, and that your interest in them should trump their interest in you. I know that’s how you feel and it’s normal… but telling professionals will not be in your favor and will likely reinforce their feeling they were right. If you really, really like Barnard and want to try and transfer later on, don’t burn your bridges by insulting them with an appeal. Your GC may ask for an assessment, express that you hope to apply as a transfer later on, so what parts of your application should you especially work on?)
What kind of schools is this (rural, but private or public? Well-resourced or not)? Where do students typically go to college? What % go to 4-year colleges? Is it a “pipeline” school or are you among the first to apply to Barnard?
Where else have you applied, and what are your safeties?
What’s your budget and do you need Financial aid, merit aid, or nothing?
Also, there are quite a few women’s colleges located in a city: Bryn Mawr (Philly), Scripps (LA), Wellesley (Boston), Agnes Scott (Atlanta).
So okay they’re not RIGHT IN THE MIDDLE of the city but they’re not rural nor isolated. I assume you’ve applied already or are in the process of finishing your essays. Your #1 priority right now must be to have your application file checked by your GC to see if there’s a glaring problem and to improve your essays so that you have a shot there.
You may also want to consider ED2, so you have 2 weeks to figure out where and why.