I have to say that it is a horrible feeling that the child must be going through when she is around dogs. Nobody would want to feel the anxiety that she feels. That being said, I would not want my child to continue feeling that way. I believe that your friend's intentions (wanting to spare her daughter those feelings) are probably good but not the best idea. Of course, my guess is that mom also has anxiety about dogs.
Unfortunately, when kids are afraid of animals for no good reason, it is because the parents have led them down that path. Children that are not exposed to and taught how to deal with dogs and other animals in a mentally healthy way will suffer over the long term. Her mom is doing her daughter a disservice to teach her that her environment will adapt to her irrational fears.
I am a dog lover and have helped my friends children overcome their fears and stop suffering from the anxiety that comes with them. It is a wonderful thing to watch a child become empowered and feel the enjoyment that comes from being around animals.
I don't believe you are wrong. You can ensure your friend that you will carefully supervise any interaction between your dog and her child. Maybe she could bring her daughter over for short visits when you can hold her child (I don't recommend mom holding her because her daughter will feed off of her anxiety) and then have the dog follow commands and do any tricks that she knows. Then have the child tell the dog to "sit". (Sometimes dogs won't listen to somebody else, but you can cleverly say "Okay, now you say 'Rover, sit!'. You have said it and the dog will listen to you, but the child says it and the dog responds so it works like magic!)
Don't force the child to pet the dog, but in your arms they usually start to feel very confident when the dog responds. You can then have the dog lay and stay at a comfortable distance while you play. Don't put the child down though until she is calm and relaxed also.
I think children feel better when they approach the dog rather than the dog coming toward them. (They probably feel a loss of control.) I have my daughters help me with the next phase. I will have the dog lay down and receive love from my kids. Allow the child to observe from a distance without pointing it out or making a fuss over it. Usually the child will calm down if they are focused on playing something with you, but you know they are still keeping an eye on what's going on and if it looks like the other kids are having fun they usually get curious and will move in closer for a better look.
See if you can get your friend to let you "borrow" her daughter for short periods of time to do these things. Maybe your friend should try these things with you, too, (Minus you picking her up! lol) without her daughter present.
In the end your friend will be happy that she has helped her daughter to not carry these awful feelings with her like baggage for the rest of her life!
If her daughter does come over for a longer period, you could try some of these things during part of the time and put the dog away part of the time if her anxiety is too high.
Here's to raising happy, well-adjusted kids to the best of our ability! (I think they forgot to give me the owner's manual when I had kids!)
I am sorry this is so long!
Best wishes to you all, L.