Get a second opinion from another vet. As them about discoid lupus erythematosus.
Ghilly
If you are sure this isn't ringworm (there would be usually some redness and possibly itching associated with the circular patches where there is hair-loss) -it may not be a food allergy. It could be that it still IS an allergy of some kind, and usually the thing you wouldn't think of can be the culprit...eg: traces of washing powder or fabric softener in the dog's bedding/lawn feed/household cleaning products/air fresheners....etc It is also possible he might have an allergy for something like perfume or deodorant which someone in your house uses?
It might be good to try completely eliminating anything you suspect in his environment, see if that makes any difference, if not, move on to another thing, and so on. There are so many chemicals in our environment. Many of them we have no reaction to, and so wouldn't suspect, so you will have to be a bit of a detective.
I might sound a bit "grunge" but for my dog's sake I only use cleaning products when I absolutely have to. Otherwise I use plain water, or plain soap and water, even around the house. Vinegar cleans things quite well, as does lemon juice. I never use air fresheners, but add a few drops of lavender oil (pure) to water in a spray bottle, and use that when or if really necessary. Even though lavender is safe for her I don't spray even that in the air when she is actually in the room!
I always remember that a dog's-eye-view of their world isn't the same as ours. They are nearer to the floor.... likely to lick at their feet (which come in contact with the floor) are likely to lick at bedding when they are licking their feet and legs, etc.
It is also possible that this could be an auto-immune response. Is there -or was there ever, any connection between the onset of these symptoms and having him vaccinated?