It sounds like a tricky dilemma, trying to make the transgendered details of that character subtle, especially when transsexualism is supposed to be a hidden/personal detail from the outside. I can only help you with so much since I'm unsure what the main plot is about or what sort of timeline/social setting I should expect asides from what you've already stated.
Chances are, if your main plot is gripping enough, the fact that one character in your story is trans shouldn't honestly take much away from the plot. There's several factors to consider and think about when approaching how much to reveal about this character, though;
1) Would it change much of the main plot, or is there a turning point in the story where the character being transgendered matters to some extent in the main plot?
2) Would it change the outcome of how major characters would react or interact with her?
3) Are you outing their transgendered status within the plot, amongst other characters, or only letting readers know?
4) What does the transgendered person do to contribute to the plot? What defines them as a significant character apart from being trans?
Special emphasis on four.. because characters shouldn't be especially significant only because of 'something' they are, they should be significant because of what they do in spite of what they might be.
Something that can help is thinking from it from the transgendered person's perspective, if thinking about it from the general scope of the entire story isn't helping. What kind of a background do they have, and what do they do to pass? Do any of those details matter to the main story? How do they contribute to the story that has nothing to do with their ->-bleeped-<-?
Depending on what details you've established for that transgendered character, and how congruent it is to the main plot; you can decide on how open they are about their past or presenting their demeanor to the main character(s). The trick here is identifying what details would help boost your primary story and the development of your main character. If, for example, the transgender has a traumatic past with her family but explaining the entirety of it doesn't add to your plot in some way, don't include it in your story. On the contrary, if explaining that past helps the main character reach some sort of understanding with transgendered people and develops them as a person or is a key plot point later on, then it could hold importance, for sure.
Besides, when they are transgendered in the first place, I doubt that the character wants to make it clear that they are trans, as they wish to blend into society living as their true gender.
Knowing that you are writing this in third-person limited voice, there will be only so many details that the audience will find out about. Perhaps the main character has a keen observation and detects that there is something off about this trans character and makes a couple of guesses as to why. ["Perhaps she is transgendered? Something about her mannerisms is... etc."]. That would probably be a good subtle way to get it across to readers without ratting them out to the audience or blatantly stating what she is.
Another suggestion is to think of specific/unique personal morals, gimmicks, or personality quirks that the trans person may have in the present as a result of being trans or going through a hard life growing up, rather than directly stating it. Maybe they're extremely non discriminating/judgmental towards other people when it comes to gender, or have a profound hatred of prejudiced people. Maybe certain things she does in terms of action will give away that she may not have always been a woman.
Good luck, I hope you figure out your plot development and characters. Although characters are highly important to the story, knowing how to write for them so the story doesn't lose focus is equally important. Ultimately, the key is knowing how to balance and tie in all these details together without overwhelming the reader.
