Having been full-time at work, in a place where nobody knows I'm trans, for a couple of months now...
No. I would not want such a bathroom. As soon as I would be required to use a unisex bathroom instead of the women's room, people would immediately stop seeing me as a woman, and just start seeing me as an "other."
In a society where gender identity wasn't such a big freaking deal, where trans women were seen as being just as female as cis women, I would be more okay with it. (Granted, in such a world why would we need a third bathroom anyway?) But just being realistic here, it isn't happening any time soon. And I refuse to be seen and treated as an "other" by society. Especially since I've long since been accepted as being just another woman. I wouldn't give that up for anything.
If society really wants to help trans people, it doesn't need a third bathroom, it needs to quit making this bathroom thing such a big deal in the first place. Just let people use the bathroom congruent with what they identify with, accept them as a valid member of that identity gender whether they pass or not, and don't be so defensive about it.
Maybe non-binaries would appreciate a third non-gendered bathroom as an option, but most of us don't need it, we just need the freedom to use the gendered bathroom that we identify with, and a segregated "trans" bathroom would only set us back.