As a behaviorist my professional advice is for her to get the dog used to the chickens and train the dog to respond to the "quiet" command. It will not only help in this situation but many others. She could find a local trainer to assist her in doing that. I teach that to all my dogs as I want them to bark if something strange is happening (why I do NOT recommend shock collars for this type of probelm) but also to let them know I am aware of the situation and they can now stop setting off the "alarm." Like many other posters mentioned it really is her problem but in order to live peacefully in a city or suburb we need to work together. Starting a positive dialog and asking her for suggestions as well as offering your own is a good way to start. I would also remind her gently that your chickens are as welcome as her dog in Seattle but her dog is the one disturbing her other neighbors. Do offer to help ease the situation so she doesn't get defensive (to keep the peace). Doing sneaky things such as using cayenne pepper can get you into trouble and some dogs as well as people are allergic to it--including myself. Fresh eggs once you get some is a good peace keeper as another poster mentioned as well.
Best of luck!!
Best of luck!!
