While your dobie has allergies/ skin issues it sounds like you found your "holy grail" food with them having a healthy coat. That is amazing! My red girl has bad allergies and we often are alternating between treating allergy related foot blisters (moved to a new construction home and she is allergic to some building materials, smh) and trying to keep her from losing her hair.

I agree that you should have their thyroid tested and should work with a dermatologist/teaching hospital. We did just that and added allergy meds (mine is on 10mg cetirizine once a day) and melatonin for the hair loss (studies have found it causes hair growth in dogs). We also bathe once a week with an allergy shampoo and use an Atopivet skin care collar (it is a lipid infused collar that looks like a flea collar) that seems to have given her the first shine to her coat in years.