The problem with that approach is women don't feel comfortable using the bathroom with a man, regardless of what he claims to be.
So who's rights are more important? The born women's or the born man's?
Example: https://twitter.com/JayShef/status/712845760287494144/photo/...
Frankly, while I like being comfortable, one should always think twice before passing laws purely for the purpose of increasing comfort. (Particularly in this case, where the discomfort appears to be far more theoretical than common). When I think of the society my grandparents, or even, heck, my PARENTS found "comfortable", I would feel very sad to live there.
Ultimately what makes someone comfortable or uncomfortable in a bathroom is the gender that they perceive others to be. They will usually not know what another person has between their legs, or what gender is written on a person's birth certificate. They just know what they look like and how they behave. And I'd bet that these criteria will better track a person's subjective gender identity than anything else. This is especially likely when one considers the fact that transgender people, like anyone else, will not want to make other people in the bathroom uncomfortable--or to be made uncomfortable themselves.
Edit: Here's another thing to think about. If we're really just worried about who is more uncomfortable, remember that for most third parties in the bathroom, this discomfort will be mild and fleeting. "That was weird, that women kinda looked like a man." But for the transgender person, the discomfort will be deeply felt and repeated daily for the rest of their lives. (Until, that is, they move to a better state.)
Or, more realistically, if your presented gender doesn't match your birth sex, or if you're "weird" and someone feels like challenging you under this law, you now don't have the right to use a public restroom. Just be honest and admit this is your goal instead of cherry-picking a single scenario.