I have a suggestion that may help given a little time. Try mixing 1 tsp of coconut oil into her food. My dogs have always liked the taste so they eat it right up. I also give them a tablespoon of plain or vanilla yogurt in the morning , on top of the dry food. They lick it right up, especially if its vanilla, plus thats excellent for their digestive tracts. I can only speak from personal experience. The dogs I have now don't suffer from this problem, but I had a Shih Tzu, Maggie who was tested for allergies, etc..all the same things you said. We live in NJ and the water is hard everywhere, most homes have water softeners, so that was also ruled out as the culprit. I think Maggie was just prone to dry skin. The coconut oil worked wonders. We use 100% organic. It lasts a long time, cause it doesn't take much. I'd say you will notice a difference in a few weeks, but the improvement will start faster then that. Worth a try
Good luck.