Earlier quoted context omitted.
The problem with that approach is that not all transgender people elect to have operations, so making that the minimum bar is unreasonable. A much more reasonable (non-fear mongering) approach is to use the gender the person identifies with. I don't know how you set that bar, maybe it's obvious, maybe it requires a note from a physician or similar. I think it would be fairly obvious to discern someone who's actually…
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?
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.)