I think it is something to do with food -if he has always done it. You sound like you know how to care for his coat-and rinse him well.
doesn't hurt to try something else -for diet. Raw does sound good --from reading I can see that takes time too but results are worth it according to waht I have read. haven't tried myself yet//did talk to butcher about some bones. In my goal to replace rawhide I read that big beef bones are good for dogs.
if you are trying another kibble-there are some good ones. Over a week mix 1 new to 3 parts old -by third day half and half and by 7th day-totally new diet.
I f you don't they could get problems with change. And you don't need him with the runs or throwing up.
So slow change is key-and simple -introduce one food at a time. then when stable and well on main diet-watch for any reactions to any additions. It is usually in a day that Kobe shows if he can eat it. My dad is bad cause he likes to share with Kobe then I have a gassy dog all the way home so results can be quick. Thank goodness no runs in the car. Told my dad kobe can't visit if he would respect Kobes diet //cause Kobe suffers for it and i do not like that at all!!
kobe too was allergic to what he was eating when he came to us. he was so itchy-the vet thought he had walking dandruff(which requires meds) but it wasn't-simple test said so.
so we changed him to medical dry food. I bought it right then and there at the vet. They did not push on me -just said that it was a high end kibble he was eating but to try another to see if it helped.
Ichiness stopped in a week and in a month his coat was clear of white flakes and he looks like he is polished once a day-

he is shiny.
I may change him again-I am just checking out different kibble mentioned on here somewhere.
I did try some wet food with his dry and anything beef seems to give him the runs. He likes pixels and freeze dried chicken or liver. I didn't know about rawhide being risky and he loved the flat ones---. so I stopped giving him the rawhide chips and bought some dried chicken.
It is supposed to be real meat and a chew for him. A whole one -which was amost the size of a strip of bacon was too much so cut in half was better.
with Kobe its gas we don't want to encourge

and we have that under control-as long as i watch every reaction with any new food.