It is normal for kids and even sometimes adults to say such things. I agree that the reason for this is that they don't know how to express their anger in a more productive way and that telling them alternatives at the same time as making it no big deal is the best way to handle it. Saying these words is not being a bully. But I don't know of any way to convince your neighbor of this unless she is willing to discuss it with an open mind.
I've heard my granddaughter say this to a couple of friends during kindergarten and first grade. She said it fairly often to her "best" friend whom she calls her adopted sister or cousin because the mothers are good friends. In all of the situations I've seen the kids response to "you're not my friend andymore" is to walk away. I have playground duty and I see them playing together within 10 minutes or so. Walking away at the moment seems logical to me. It does hurt to hear those words and by walking away they are providing a logical consequence to the other's words.
Sometimes my granddaughter is hurt when someone says it to her which gives me an opportunity to suggest that now that she knows how it hurts perhaps she could work on not saying it herself and then later talking about other ways to express her emotions. It seems that her "best friend" and my granddaughter do fight less often but I suspect that is in part because they are more mature. They're entering second grade in the fall. Just as kids go through physical stages of growth from being unable to even turn over to running they go through emotional stages of growth.
Perhaps a book talking about the emotional growth of children might help your friend. I don't know the one you ordered but it sounds like it might help your friend to understand the dynamics and be able to reassure her daughter that the "I hate you" or "you're not my friend anymore" is only temporary.
Also that by reassuring her daughter and discussing ways to handle this she is teaching her how to deal with her own emotions.
Perhaps your friend sees the world more in black and white than shades of gray and she herself doesn't know how to handle this sort of situation. If so she is bound to be defensive and unwilling to see a different view. This means that your approach needs to be more indirect than talking directly with her. I'd suggest that after you've read the book that you say in passing that she has raised your concern about this issue and that you found this book to be helpful. Don't state your case or try to defend your daughter or even your own reactions to it.
One theory that I have found useful even tho I can't always act upon it is that the more we fight against something the stronger that something becomes. Often when we back off it allows the other person space in which to review her own beliefs.
I think, in this situation, I would tell my daughter that this mother's view is another way of looking at the words and that even tho you don't agree with it you will honor it. I expect that within a short time the two girls are playing together again.
It just struck me that your friend is responding to the "you're not my friend anymore" statement by also saying "you're not my friend anymore" when she says that her daughter can't play with your daughter. Friendship also involves both fun times and difficult times. It sounds like the two of you are remaining friends and if so the two of you are providing her an example of how to deal with the difficult times. Perhaps discussing this idea, which is less judgemental and in fact gives a positive message to her, might work. First tell her that even tho you disagree you will accept her boundary.
Also explain the issue of differing ways of looking at things with your daughter. I realize that I'm assuming that the girls are school age. But even if they are toddlers a reassuring message to your daughter would be helpful for her self-esteem.
I'm seeing a broader scope of ways to handle this as I type. My first reaction, if I was in your position, would be to be defensive myself and angry because this seems so unfair to my daughter. My second step would be to "fix" the situation by trying to change my friend's belief. Over the years I've finally realized that I can't "fix" most situations by trying to change the other person. The only person I have any control over is myself. And some situations do provide me with the opportunity to teach children how to feel powerful by accepting the other's viewpoint no matter how unreasonable it is by figuring out how to take care of one's self instead of focusing on changing the other person. This is a very hard concept to grasp and act upon. It's taken me years of learning about boundaries and how to make my own boundaries to be able to handle conflicts in this way.
And I still am not able to do it at least half the time because my emotional reaction is to "fight" back by showing them that their belief or action is erroneous and/or unfair. I usually do this in a teaching mode, intellectual with my emotions "in control," (I was a high school teacher many many years ago) Even tho I think I'm being reasonable it usually doesn't work unless I first acknowledge the other's feelings and beliefs as being reasonable from their viewpoint and even tho I don't agree with them I accept their right to hold them. It may take days or even weeks of accepting their beliefs in this manner before they can relax enough to listen to your beliefs. I find it difficult to be patient.
So, in this situation, I would tell my friend just that. And I would talk with my daughter about ways to handle the situation. If she's a preschooler, I'd just say something to the effect that not being allowed to play with her friend does hurt and reassure her that she is a good child in spite of the angry words. Then when emotions have cooled down and your friend seems amenable to a further discussion or an acceptance of loaning her the book do that.
I hope this helps. I've used too many words but I don't have time to edit it. My heart goes out to you and your daughter.