First, it’s “transgender”, not “transgendered”. It is an adjective, not a verb.
Second, the military isn’t going to let people claim to be transgender as a lark or as a means of bypassing regulations. As with many other things in the military, there are more stringent requirements than exist in the civilian world. And even in the civilian world, there are legal standards for establishing a legal gender identity other than that assigned at birth; check with your local DMV or your state’s office for birth certificates, and you’ll see that, while at the Federal level there’s requirements from the State Dept. for establishing such in order to change your passport.
Not jumping through the hoops doesn’t mean that the person isn’t transgender; it simply means that they haven’t put forth the due effort to establish that to the satisfaction of the relevant authorities.