Or is this just a case of the language not offering enough choices to cover all situations?
Pretty much. The character I have in mind would have begun transition already with some other characters already referring to him/her as "she." It's kind of the situation I'm in now; write what you know I guess, just as long as it doesn't turn into some kind of self-insert fantasy.
One of the things Goldsmith does in Genma's Daughter, which is a Ranma 1/2 fanfic, is to refer to Ranma as he or she depending on which form he/she's in but always as Ranma until he/she decides to lock the Juusenkyo curse and go full time living as a woman for a few weeks. At that point, the author begins referring to the character as Ranko. Not to give spoilers if you haven't read it, but there are only a few times when Ranko is in male form after that, and Goldsmith reliably refers to the character with the name "Ranko" and the pronoun "he."
In the non-fiction book Whipping Girl, Serano observes that pronouns are often a matter of presumption and contextual guessing that one has to make within the first few moments of meeting somebody. After all, we generally don't go around performing a "Crocodile Dundee" manuever to verify which body parts a stranger has between their legs before we decide whether to use "he" or "she" when referring to that person. Checking others' genitals would also lead to ambiguity in the case of trans women who have the female gender and a woman's name on their driver's license but have not yet undergone (or does not desire) surgery.
One might argue that the state of one's genitals is of utmost importance, which is one thing that feminists and right wing authoritarians can both agree upon. However, because we do not check others' genitals regularly, it becomes a moot point unless there's romance involved. The zombie flic Wild Zero presents one such situation.
As Serano also notes, sometimes we get it wrong. As an example, I used to work with a perfectly cisgendered man when I used to do fast food who would often be ma'amed when talking to customers over the drive-through speaker because he had a feminine way of speaking. Sometimes when I'm very tired, I'll get ma'amed by strangers who have only my appearance to go on when clearly presenting as male simply because when I'm tired, I forget to use masculine body language. So, it's not as straight-forward of a process as we'd like to imagine it is.
I feel that getting the language thing down would be one of the challenges of presenting something like that to a wider audience. One solution might be to write from first person, but it would be more interesting to use third person omniscient so I can use the same technique as Goldsmith. I feel that the way other characters use he or she can be an effective way to demonstrate what other characters think of the protagonist's gender---the narrator's choice of pronoun being dependent on whether the character is presenting as male or female---, and which particular name the narrator uses to refer to the protagonist can demonstrate the state of his/her own identity at that point in the story.