I think she's a member of the Choir alright, but may have either been from Cainhurst or is a spy from Cainhurst. I don't think OG Iosefka was from the Choir, I think they just used her impostor model at the door to save time since you can never actually physically see regular Iosefka without tricking the game. But yeah, impostor Iosefka has gotta be Choir; her outfit, her experiments, her magic, it just fits too well. The Cainhurst stuff is the only kink and I don't think that explaining that or adding it to her character eliminates her being a full blown Choir member.
I never thought about the possibility of her being both from Cainhurst and the Choir, you're definitely right that's very possible, especially with the whole spying thing that goes on between Mensis and the Choir.
But I just assumed she got the outfit and hunter tools from Iosefka herself, before turning her into a celestial mob. Explains Iosefka's outfit with more than just "videogames".
Though only "videogames" would explain why they look exactly the same.
Ok this made me double check and I am lost now.
Wiki says that fake Iosefka only drops the cord if you kill her after killing Rom. Which suggests she used your incursion into Byrgenwerth to take the cord.
Which means the cord itself is not needed to create Celestials? But the church is also creating celestials, which suggest her to be part of the choir.
But on the other hand, at that point Iosefka seems to be pregnant, and killing her drops the cord. Which has a description that makes it seem like it was Willem's. But since she is pregnant, then maybe it was her cord, and the description is just for flavor??
And I cannot make sense of her Cainhurst connection. Other than you having a Cainhurst summon, which may be the reason she is friendly to you, she thinks you were sent by Cainhurst.
Her Cainhurst connection comes from the fact that she gives you Numbing Mist for every person you send there, and Numbing Mist is a secret recipe guarded by Cainhurst nobles.
Every carrier of a child of blood seems to necessarily be connected to the corrupted blood of Cainhurst, possibly related to Yharnam herself in some way. Arianna wears the Noble Dress, which is also from Cainhurst, and her blood's description says it's very similar to the forbidden blood, or something of the sort. And the Tsundere Queen is just from Cainhurst, period, no need to speculate.