I had the same trouble with one of my dogs. Every summer she got yeast infections in her feet. She'd cool off in the pool, and since she had big hairy toe pads, they harbored moisture and made a perfect environment for fungus. The stench was horrible!
My vet recommended Dermachlor Flush Plus, and it was a godsend. It comes in a bottle and I'd squirt it on her toe pads and in between her toes twice a day for 3 days. Fungus gone. It also immediately got rid of the nasty smell, and since she didn't like the taste, she stopped chewing.
Of course, during the hot summer months, the fungus would come back in about 6 weeks, but at least I could recognize the signs and immediately treat it. It got to where she'd see the blue bottle appear and willing stand still for treatment. If you've ever seen a horse having his hooves cleaned, I used the same procedure. My dog was much more comfortable standing on all four legs and giving me one paw at a time. Ask your vet if Dermachlor might help. It's not expensive and hopefully will help if it's a simple yeast or other fungal infection.
Dogs chew their feet because the feet are itchy. Causes for itchy feet include allergies (ie to pollen, grass, or food), bacterial or yeast skin infections, and skin mites/parasites. The bad odor suggests that infection is present, and veterinary exam is needed to identify the type of infection so that appropriate medications can be prescribed. The bad breath may also be due to dental disease. and your veterinarian can also examine your dog's teeth to see if there is periodontal disease; a dental cleaning and brushing his teeth regularly are usually very helpful. If your veterinarian has already examined your dog and has not been able to get to the root of the problem, then referral to a veterinary dermatologist may be helpful (www.acvd.org).
Kimberly Coyner, DVM DACVD