Dog professional here..
DO NOT GET A DOG...
Childrens normal behaviour (dashing about, squealing, being wobbly, unpredictable) is very upsetting for dogs, they are likely to bark, nip, growl etc.
Childrens behaviour when scared - panicking, squealing, dashing about, pushing away etc... will REALLY upset a dog.
Puppies run about biting barking and jumping until they are mature enough, and have learned alternative behaviours.
Adult rescue dogs are to some degree an unknown quantity, and may not react well to your childs behaviour.
Instead... speak to local trainers, find some starting a puppy class where you can sit in a corner, with a barrier, so your daughter can WATCH puppies for a short while each week.
Build up from there to watching adult dog classes, perhaps seeing steady dogs at closer quarters, eventually perhaps touching a dog.
Then when the fear is addressed, consider getting a dog if YOU want one, and, if your daughter wants one.