I would most certainly get a 2nd opinion.
I'm not a vet but we had 2 dogs who had skin allergies. It looked similar to how your dogs skin looks. One of our dogs had it because she was allergic to red meat. So we had to feed her food that had chicken or fish, no red meat.
Our other dog had skin allergies that were awful. She was allergic to something outside. Our vet told us to give her benadryl. It helped to a small extent, mostly to make us feel like we were doing something, but wasn't nearly enough to help her.
So we took her to a homeopathic vet and did research online. Long story short we had a long list of daily routines to help keep her skin clean and that made a huge difference in her allergies for the better.
But you never know what it could be with your sweet dog, maybe not allergies. I would definitely get a 2nd opinion and make sure that vet takes it seriously.
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