OP, this woman extracting information from your daughter and then pushing in, is a complete NoNo.
She clearly does not care about your child as she is focused on what works for her and or her daughter.
She is trampling on your daughters privacy and I wouldn't tolerate it for a minute.
I would push back really hard and whether the friendship survived would not bother me at all.
Infact I would think space would be best.
I cannot bear behaviour like that and you are doing your daughter a huge disservice by allowing this to continue.
I would tell her straight that YOUR daughter is entitled to her privacy to make other friendships and her quizzing her for information is deeply inappropriate and intrusive.
Tell her that if it continues you will cease contact completely.
She is not your friend and she is doing her daughter no favours whatsoever.
She needs to be told back off.
Personally her behaviour would give me the Ick and I wouldn't have any interest in being involved with someone so controlling.
She is actively interfering in your daughters life and you are allowing it.
Model firm boundaries for your daughter and explain to your daughter why.