If there is a possible sensitivity to an ingredient in a dog's diet this can manifest as the dog being itchy, and an overgrowth of yeast levels which would usually be kept in check.
Reggie you mentioned reactions on the previous diet your dog was fed, and also reactions to the more recent diet, which leads me to prompt that it may be wise to look into what the ingredients are in the foods, in case she has a food sensitivity. And by doing so you'd be trying to work out if she may be reacting to a particular ingredient.
If she is food sensitive then naturally it follows that by eliminating the 'culprit' ingredient(s) from her diet she should be able to lose weight comfortably, without reacting as she currently does.
Looking at Nutriment Low Purine & Phosphorus (food 1) the ingredients are: beef, chicken, salmon oil, wheatgerm, carrots, broccoli, butternut squash, bilberry, coconut oil, sesame seeds, spirulina, kelp.
The ingredients of Nutriment Turkey Chubb (food 2) are the same as Purine except chicken is substituted with turkey.
Natural Instinct Special Diet (food 3) ingredients are: beef, chicken, salmon oil, carrot, apple, butternut squash, spinach, kelp, brewer's yeast
Logically you can see that there are several ingredients common to all the foods, salmon oil for example. There are also ingredients that are specific only to one of the foods eg. brewers yeast is only in food 3, turkey is only in food 2. And there are ingredients that are in two of the foods eg chicken is in foods 1 and 3.
If you wanted to also compare these ingredients with those in her previous diet you may get an even clearer pattern as to which ingredient(s) may be causing her to react as she does.