Yeah, there's likely some kind of workarounds, but also, the best solution is everybody banding together to make it impossible for management to ignore (this seems unlikely to happen here though.)

Although honestly, I would be tempted to just start reminding coworker to bring her parka before she goes, and acquire one cheap to loan if needed. It would solve the joke problem with a taste of her own medicine while acknowledging the discomfort that's inspiring it, and also keep her warm in the bathroom. (For reasons that don't need exploring at this juncture, we have a small collection of unclaimed real fur coats at my workplace, and yes, they do get worn and loaned out as a visible protest some days.)