so the director is not trans? they're just doing that for their dad? that chapter was kind...

I think they might be because they said to the brother they dress like that because they want to

I believe he meant that he dresses like and pretends to be his sister for his father’s sake of his own volition, not because he likes to dress like a woman.

So I've thought about it for a bit and I think I figured out what's going on.
The director is trans. They are female presenting their workplace which isn't something they need to do to convince their dad to take a pill. They also asked that very strange "how would you feel if your lover said he wanted to be a girl" question which doesn't make any sense unless the director is trans. They really do want to be a woman but social expectations are holding them back.
However, dressing as their sister obviously made them feel very uncomfortable. But they still did it willingly and said as much. But why though? Because they feel like that's the only way to dress as their preferred gender in front of their dad and get a positive response. That also perfectly explains the apology to the sister after (I'm sorry I'm using you to get affection from father).

omg you got this spot on