There are many, many straight women parading around these days calling themselves "trans men" without ever taking testosterone (and I think it takes a while for testosterone to stop the period completely anyway), just like there are a whole slew of troons who will never take estrogen. Those women all have periods, but I seriously doubt a single one of them would ever use the men's bathroom out of (rightful) fear of male violence. Patterns show that most trans-identified females prefer to use the women's bathroom, regardless of hormone status, just like they prefer to be on women's sports teams and women's prisons. It's one of the many ways the trans-identified male platform of "everyone should use facilities/services matching their gender identity" screws things up for females who actually are at risk of violence, rape and death should they try to force themselves into male spaces.
Anyway, yeah, I don't think anybody uses those dispensers anyway. Every one I've seen has had shit quality products that could be one-upped by loads of toilet paper lol. I've never seen a single woman ever use one.