@Kaarme
I don't think she was going to tell everyone, they are family and she has been teasing him all the time. It seems to me that you're projecting a lot into a simple situation. It's mostly because now that they are alone the nurse can reveal her partner is trans and thats why Kei couldn't remember.
But again if Kei doesn't want to be seen then he shouldn't take a job where he is in public, he had the choice to say no, or quit, nobody is forcing him to do anything. Don't wanna be seen in public don't work a public job simple.