Starting with your utterly denigrating and offensive use of the term "he/she", which the Associated Press Style Guide says you don't do. The person is living as a woman, use female pronouns. Anything else is disrespectful in the extreme. A transsexual - even pre-operatively - is not a "he/she" - that's a crude term that merely demonstrates your ignorance of the subject.
On the other hand, having this same fully endowed person in a girls-only changing room is almost certainly something the women wouldn't want. In fact, the consensus of the admittedly modest sample I surveyed was that it would "creep me out."
Okay, I can - to some extent - accept that reality. But, a MTF transsexual can no more use the men's changing rooms than the women's in that circumstance. (In fact, arguably, the men's changing facility is potentially quite dangerous to the transsexual)
But, the question has to be asked - are there single person washrooms that she could use? Would it be acceptable for her to use the shower area if she wore a bathing suit to cover up the bits everyone is so freaked out about? (Do I really need to explain that by the time someone transitions to living full time, chances are the bits between the legs aren't functioning the same way they did when the body had a full dose of testosterone in it?)
So, while transsexuals have rights, they are not insuperable rights: get the operation done, then join the gym. It won't hurt you to wait.
No. Wrong. Dead wrong in fact. Using that logic, it becomes even easier to ban transsexuals from using the gender-correct bathrooms in public places. Not every locale has single occupancy bathrooms, and transsexuals, like every body else, need to pee from time to time.
What the club owner's mistake was is a failure to communicate. When you "put someone off" like that, you had better make sure you are communicating your intentions and progress with them. Otherwise, they will quite rightly assume that you are just hoping that they will go away.
Most transsexuals will live with some kind of compromise solution as I suggested above. It's not like they are unaware of the impact that they have on those around them. (They may not be happy about it, but they'll live with it) Telling them to go "hide in the closet until they are post-op is not unrealistic, it's stupid.
It is not creating a better world, however, when self-absorbed people put other people in a bind to advance what they see as a right, and HRCs abet them.
You sir, have just passed beyond offensive. How is trying to live a perfectly conventional life (including exercise) "self-absorbed"???