Responses
It's a patriarcal social class. She cannot ask for custody. She is a noble but no home or power (remember she and her sister got thrown out of the their own house). Asking for blonde duke for protection is the best choice she have.
If you are reading the tone of the conversation between her and the king, it's not friendly, it's already decided. Even if she asked the king, nothing will change, she will only anger the king (it's not one of those scene the FL can do what she wants and the king will follow her because she find her interesting)
I mean fl is also dumb. She had her chances to the king. I was disappointed with her terms. She could just say that if they succeed (they will as it is the plot) the subjugation or something, she will have the full custody of the child. She can ask for that anyway. But instead she proposes dumb suggestions which the blonde duke or whatever his name is already provided the protection. If I were the fl, I'll take the kid with me.