I just toss this out for you to think about, not as advice.
You're right, she's overstepping her bounds in asking and also in discussing with your DD who picked her up.
But if you haven't filed yet and your custody matters haven't yet been determined by the Court, I wonder if it would do you well to let your WS pick her up after school when you can't for a couple of reasons.
#1, it shows the court you're willing to be flexible, to be a cooperative coparent and to do what's in the best interest of your DD; it's also showing your DD that you are focussed on her, her happiness and well-being and you are going to be the rock solid parent who stays even keel and is reasonable.
#2, I'm guessing it's very difficult for your WS to not see her daughter after school because, in spite of her terrible and hateful behavior, she still has maternal hormones flowing. So if you allow the after school pickups to allow her a bit of time with DD, it might help her realize that she doesn't have to be vicious in the custody proposal.
#3, it might also be good for your DD to see her mother during your week; she's having to make a big adjustment in her life and it might do her a world of good to know that you and her mother will use some flexibility in order for her to be happy. In fact, if it can be worked out easily, maybe WS would agree to you seeing DD for awhile during her week with WS?
And, keep in mind, this might be a temporary thing because I assume your WS will be getting a job? And once she's working, she won't be sitting around ruminating on stuff so much.
And, of course, document everything.
As an aside, I would encourage you to never talk to your WS in front of your DD. I also encourage you to not answer the phone when she calls - let her leave a message and that gives you a little time to think about what she's got to say before you call her back. Even better, confine your communications to email and text; that way, there's a record of what you both said and/or agreed to. And, not hearing her voice has the added benefit of you being able to heal quicker.
A lot of families use a parenting app to deal with all the matters pertaining to the kids. It might be worth looking into.
I'm sorry you're going through all this. It's the pits and there's no easy answers out there.
BW, was 67; now 73; M 45 yrs., T 49 yrs.
DDay#1, 1982; DDay#2, May, 2017. D July, 2017