Sounds like her biggest concern is that her garage is being broken into.
Why don't you discuss the noise issue with her and offer to help her pay for motion detector lights, alarms system, etc. The cost for such a system is not much and may be less that she'll pay for a good dog and training.
You can tie in a motion detector light to an alarm, a video recorder, an alarm monitoring system, a phone dialer....
I installed a motion detector light for a friend on the back wall of his garage, and ever since he's had no problems with intruders.
On the issue of the dog, I think that dogs that bark all day or all night are not happy dogs, and you may want to point out to your neighbor that she may be being a bit cruel to train a dog that way...