Yeah, it's hard to blame the dog when it was trained to use a (stuffed toy) chicken as a squeaky toy to play with for fun. Probably very confusing for the dog, too. Sounds like you are doing all you can for your injured pullet.

FWIW, I don't treat my chickens as pets. If anyone of them gets sick or hurt, I'll do my best to try to nurse it back to health, but, I won't bring any of my chickens to our local vets. As I have said a number of times, I can replace my entire backyard flock a number of times over for the cost of a single vet visit. I know a number of people on the BYC community treat their chickens as pets, and it is worth it to them to take a sick or injured chicken to the vet, but I just want to add my sentiment that you don't have to feel guilty about NOT taking a sick or injured chicken to the vet. Over the years, I have learned how to save some of my chickens when sick or injured, but not all. I am willing to accept a loss here or there knowing that ultimately, chickens have a short life span, and can be replaced if needed. That may sound harsh, but that attitude works for me and has allowed me financially over 40+ years of having and caring for small animals.
In any case, I hope your chicken recovers from her wounds. Additionally, it might be a good time to (re)think the safety of your chickens in your setup. I keep my chickens in a fenced chicken run where they are safe from a neighborhood dog wandering through my property. So far, 3+ years, I have had a number of dogs sniffing around my chicken coop and run, but no attacks. The fencing has kept them on the outside.