I see no reason why not. Rape is not even an issue. I see no harm from it. Even if it makes someone uncomfortable that is their problem, not the transexual's. If they truly live life as the opposite sex why bother them. Using the proper restroom is an important part of transition.
The reason we have separate bathrooms is because of the psychical differences between the sexes, not the social differences, so it does not natter what you socially identify as, if you are physically a man you must use the bathroom designed for physical men. What happens if a 5 year old girl wants to go swimming? What happens if this girl, who has not seen a penis and would be freaked out by seeing one, sees a naked man in the shower room? What about the pre teen girls who want to go swimming and don't want to see naked men in the shower room? What abut the women like my self who don't want to get dressed or shower in the same room as biological men? Why do we have to sacrifice our comfort and safety for 0.03% of the population?
Especially if they haven't had surgery or had hormones I feel they shouldn't be allowed to go in the restrooms they want. At the end of the day the other people who are in those restrooms don't know that transponder personally, so they don't know what their story is etc. All they see is a man coming into the ladies room and a lady coming in the men's room. How is a complete stranger suppose to know that you are a transponder unless you have a sticker on your shirt or a tattoo saying hey I'm transponder. To stop confusion and being problematic just go into the bathroom that society identifies you as.
If transgender people are allowed. A physically male teen could use the female locker rooms and showers or a physically female teen could use the male locker room and showers. Obviously, this would make things even worse for all parties involved. Tho seeing the opposite sex may not bother the transgender person, being stared at probably would. Further, it would be disruptive to the others using the facilities as you could imagine.
Tho that was dealing with teens, there would likely be the same/similar issue with adults. The best solution is for those with male parts to use male facilities and those with female parts to use female facilities. After all, they have seen their own parts so seeing others same parts would not make any difference.
Transgender people should not be allowed to use restrooms and showers that differs from their birth sex. You cannot change mother nature. You can live like the gender that you weren't born as but it doesn't make you that gender. It is time we stop enabling these people that obviously have mental issues.