it just sits real weird for the Director to say that.
I don't think it's a Japanese culture thing so much as an entertainment industry thing. People use diminutive terms for young women in professional settings rather too often as it is, and in the context of idoldom and curating a "familial" environment for these girls it doesn't seem super unusual. Obviously I'm unaware what the original term used was, though.
So, Miyako can't tell that manager to take a hike, because ... ?
Because she's considering that the director's plan may be the best option. She doesn't want to create difficulties for her friends' careers and she doesn't want to risk exposing her partner to invasive media attention.