Your situation is strange. I can only comment on another type of cyst problem though. I know a woman that was prone to cysts, generally all over, non-bartholin. Her doctor told her to stop drinking caffeine and her perpetual cyst problem would stop. Turns out, the cysts have not come back.
Is your bartholin's gland blocked prior to the cysts forming? Do you notice any changes around the area after consuming the stated foods that cause the cysts within any time span post-consumption to the cysts?
What about taking warm baths with the area exposed to the warmth to help release and clean out the area?
I've had a splinter underneath my fingernail for a week (yes, it hurt horribly bad) and a bath softened the tissue and lubricated the posterior nail that housed the splinter. It came out without any pain (unlike other tries) with little effort.