First off, why even bother with the picture? If this is all about principle, should it really matter how feminine Bridget looks? Do we start excluding validated Shriner-wannabes because they aren't pretty enough?
Second, it's called the shrine of FEMALE listeners. Now, perhaps by calling it that, Ian meant for it to be inclusive enough for folks like Bridget. However, I think it's fairly obvious that the commonly accepted meaning of FEMALE means, in general, just what Manwich said: two X chromosomes.
I also think that it's obvious that it DOESN'T mean men (especially those who are genetically and physiologically male) who dress and act like women.
Bluntly, by allowing Bridget in, you change the Shrine to something else. The title "Shrine of Female Listeners" is no longer accurate in the minds of most of the viewers. Basically, it's now a lie.
(Please save the "but femininity really means..." arguments. While Bridget is free to feel, believe, and act as he chooses, no amount of hormone implants or surgery can make him female. Can he live as a female? Yes. And he can go right on ahead with that. Can he actually, physically ever BE a female? Only if you're delusional--or have a transmogrifier--or just can't grasp the basic concept of male and female as accepted by the vast majority of societies. Truth doesn't bend, folks. If he has one X and one Y, he's a man. Period.)
-Wayne