I think that in order to use those rooms an individual should have made actual real steps to transition. Someone who hasn't taken hormones, who doesn't dress appropriate to the sex they declare, who isn't living like that sex doesn't strike me as being genuine in their transness. Declarations alone don't demonstrate reality. Changing sex requires an enormous amount of work. If someone's not going to do any of the work why should they be allowed to enjoy the fruits of the labor?
Which incidentally is a very Catholic POV: "faith alone" is just thinking nice thoughts. You've got to do something for it to count.
But that's all just my personal opinion, I have neither the means or the desire to make other people abide by it.
Fair enough.
That raises a lot of problems, though, for which there are no solutions currently on the table.
What does it mean to dress like a woman? Who gets to decide if you're femme enough to use the women's bathroom?
Who gets to see your medical records, proving a sufficient degree of transitional effort?
Why should a pre-op transsexual be barred from the bathroom their condition calls for, simply because they haven't yet had the good fortune to be two or three years farther along in their transition? Etc.
Part of the reason "just take them at their word" is being proposed is because these are very difficult questions to answer. Most of the answers end up being either intolerably intrusive, or putting too much power in the hands of the mob, or both.
On the other hand, "just take them at their word" undermines the social norms and expectations that underpin the current honor system for bathroom access. Right now, today, a woman can stand at the door of the ladies' room and legitimately block access to anyone who looks like a dude. And for the most part, if the dude tries to go in anyway, she can legitimately use force to keep him out. He can be arrested for trespassing, harassment, and probably some kind of hate crime.
"Take them at their word" does away with that. The woman in this scenario goes from being an important social gatekeeper to being the actual hate criminal. The risk, raised by Rolfe and others, is that predatory men who were previously deterred by this social gatekeeping will be emboldened to ignore the honor system and make their moves.
The risk alone is sufficient for me to conclude that we should hesitate before implementing "take them at their word". But none of the alternatives seem particularly workable either.
Your third bathroom has some advantages, in that it gives femme transwomen a safe place to go, without opening the "who gatekeeps the ladies' room" can of worms. But it's not always a practical solution (cost of retrofitting, opportunity costs for that floor space, etc.). And it's still a compromise WRT trans-identity, which doesn't really seem fair to transpeople.
As solutions go, your compromise seems like a bit of a compromise. It's not a meeting in the middle kind of solution. It's more of a stopgap while we try to figure out the more difficult problems as a society over time. It eases a bit of the pain for a little while, but the real problems are still there, and still need to be solved somehow.