I know it sounds odd, especially as your daughter has other friends and you are good about setting up play dates. As a mom of an only daughter, I too am conscious I need to set up some play dates, so I can kind of see your situation. But I've never understood the whole "play with kids in the neighborhood" thing since we have no kids her age right around us, and she does not go to the local school.
I'd advise you to get the book "Queen Bees and Wannabes" from the library. It shows how girls can get into these really dysfunctional relationships and let themselves be bullied for so many different reasons. It may not apply perfectly and the cases tend to be those of older girls, but you will find a lot in it to give you things to consider here.
I don't quite buy the "she's the only person nearby to play with" argument your daughter uses. SHE believes it sincerely, I'm sure, but since she has plenty of other activities and only one single free day each week -- she really doesn't need to play with anyone that one day; in fact I'd direct her to other activities at home because all kids need down time and alone time, and those are things they have to learn for themselves. She can't learn that if she is in a play date, activity or seeing this kid (or another neighbor) every single day. Make that one free day a day when she (after homework and chores!) can do something solo of her choosing -- paint her own nails, do a craft, write a letter, repaint an old jewelry box for her own use, etc. Get creative and encourage her to do so. Let her know this time is hers alone to do what she wants, and she does not have to say yes to anyone who wants her time just then.
She also needs to learn that real friends do not treat you like this, and you do not have to "settle" for this treatment just to have something to do one afternoon now and then. Is she in Girl Scouts? That helps build positive relationships (and if the leader is good, even girls who do not get along will learn they have to deal with each other). If not, look at the Queen Bees book for ideas about reinforcing what a real and positive friendship is like. But don't lecture her -- she may end up defending and speaking positively about the "mean" kid. Children this age will defend another child to an adult who is criticizing that child -- I know this from experience -- even if the other child is simply in the wrong or not treating them well. Don't make her defensive about this girl but do, without any reference to this girl, encourage a once-a-week special "me time" for her or a "mom and me" time at home.
If the other girl(s) call and want to play with her, she needs to know: She DOES have the option to say, "Thanks, but I can't. I'm busy." And "busy" can mean just being at home doing what she wants to do without them around.