I think you are right, she has more than one thing going on. Even on the correct dose, I still am prone to tummy trouble, dry skin, etc. but to a much lesser degree than before, so I assume the same holds true for dogs. But I think that is just a facet of what is going on with her.
I don’t always like suggesting tough love, but I think to get a handle on what is causing her issues, you need to pick a food and stick with it. I know you said she vomits if she doesn’t eat, but you also said she is overweight. So this is what I suggest; get either the LID turkey or lamb or some other LID food that doesn’t have chicken or beef and feed that, with no toppers. Also find, make, whatever treats out of the same protein source. A couple times a day, do a little training so you can give her some of her treats, enough for her to have something on her stomach, not enough to be a replacement meal. Offer her her kibble twice a day for 15 minutes then pick it up and put it away. When she has been on one food for a few weeks re evaluate how her skin is doing and decide if you need to try a different protein source to make things better. If you could get probiotic and digestive enzymes into her it would be for the best. The goal is to get her skin calmed down enough that you can start switching to other protein sources and actually tell when something bothers her.
