One other thought. Hormones.
Granted, it was a cat, but ....
Tinker started to itch. The cat literally licked and scratched every bit of fur off of every bit of body she could reach. The vet and Tinker's family tried everything. Then the vet suggested hormones. It turned out that very low doses of human birth control pills did the job. And, as it turned out, she didn't have to take them forever.
Funny story. The vet gave me a scrip for the pills and sent me to the local pharmacy. The pharmacist gave the scrip a quick look and cleared his throat. He said, "Ummm, ma'am, these aren't going to work for you. They are just not strong enough." When I could quit laughing, I told him that I didn't get meds for myself at the vet; that the birth control pills were for my cat who we suspected had a hormone imbalance (she had been neutered before she was rescued).
The pills went home and the cat stopped itching.