A trans person's identity could be worked into a story organically and other stories have done it. It also doesn't need to have anything to do with how they are born since it's only cis people who are obsessed with any changes or what they were like at birth. Most trans people prefer to not refer to how they were before because many find it be a traumatic experience and it brings up painful memories. The quickest way to piss off a trans person is to ask about surgeries, their genitals or how they were born a man or woman. Transition is a brief period in their lives and doesn't really need to ever come up. But given the media's obsession with it, it's all that the general public seems to understand since it's an alien experience. For trans people, they were always the gender they identify as. But no one believed them. Saying "birth gender" is a meaningless phrase since we were already born our correct gender, only our body and people's view of us was wrong.
Also pretending a character is trans without any mention of it is still erasure since until we hear otherwise everyone just assumes they're cis. You might as well say Lorca could be a Cylon since he hasn't said he wasn't a Cylon. It's idiotic and insulting. "Look they could be trans. But we don't want to ever mention it because we have some bullshit explanation based on it being in the future." Imagine if Geordi didn't have the VISOR and we were never told he was blind or that Data was an android since he looked and behaved like any other human. We'd lose a lot of stories.
Just because you can't understand how to do a story doesn't make it impossible. Because I can think of several possibilities.
We also didn't know that Kirk had a brother until we were told. Or that he was a father until we saw his son. We also didn't know that the Spock had two siblings and a quasi-adoptive sister until we we've been told. Did you know that Chief O'Brien was afraid of spiders when you first saw him in Encounter at Farpoint? A fictional character can be and become anything.