I have a simple perianal abscess that turned into a fistula, so I never needed gyne.
However, based on where yours is, I would keep gyne onboard (your GI would probably suggest that, too), but maybe see the GI as well *OR* see if your gyne would go ahead and get further tests to present to GI (such as an MRI; my fistula was dx via an MRI).
It's certainly not unusual for women on our site to get fistulas that run from the GI tract to the vagina, so it's certainly a possibility. Also, I know with perianal ones, either spontaneous draining or surgical draining will lead to a 50% chance of a fistula. I am one of the unlucky 50%.
Have you used sitz baths? Really really warm (almost hot) sitz baths with epsom salts may be soothing, and allow the cyst to drain on its own vs. having the physician drain it. I used a lot of them, too. It's what allowed mine to drain when it did... and boy what a relief it was when that happened!
Those types of abscesses are notoriously untouched by oral antibiotic therapy, too. I know my perianal abscess didn't budge when I took oral antibiotics.