I think the leerberg site has a bite-chart that suggests whether or not a dog that has bitten and to what extent that the dog can be worked with.
From what you posted, I think your dog is a good candidate to be worked with. It is awful that this happened. I don't know that it is the end of the world. But you have to be a LOT more careful with her, now that you know what she is capable of.
This was a fraidy dog when you got her. Has this been the case all along. Is it loud noises that make her afraid? What all makes her fearful.
A single bite on a hand, when you have a fearful dog -- I am guessing a fear-biter now. But, the thing is, she got loose from you, and fearful dogs, when they have the opportunity, generally run, when they are connected, flight is taken away, so they must fight. I don't know why the dog when it realized it was not connected, chose to confront and bite the child.
I don't like what your trainer said about it being protective. I don't think so. I do think this is fixable. Too much aggression is automatically attributed to fear, or protective/guarding, or dominance. It is extremely important that your dog is diagnosed properly so you can choose the best approach.
In the mean-time, don't carry a load of groceries and two dogs' leashes into your house. Leave the dog in the car, and take the groceries in, and then come back for the dog or vice-versa. Too many people try to do too many things at once. If I see someone with a couple of dogs and a passel of children, no way am I going to stop and try to pet the dogs, or talk to the person. No way. The person is over-loaded and doesn't know it. If one of those dogs chose to do its own thing, what would she do?
This is something about the management that you now have to consider. Your dog, whether this was a response due to something it was afraid of, or because it wanted to stop the noise, or chase away the people that don't belong, doesn't matter, it will bite. And therefore, you don't have the luxury of dropping the lead or being in a situation where you have too much in your hands.
You've had this dog for several months. I think she is your problem now. I think that the rescue probably wouldn't work with her and find her a new home, because she bit a child, and there are so many dogs out there that need homes, that haven't. It is a liability issue for them.
I think you are probably going to need to work with a behaviorist/trainer to give your girl a chance. A 3 year old dog ought to be able to pick up basic obedience really fast. Doing that with her on her own, and then together with your other dog, will help build her confidence and trust in you.
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