@fBoyle-- I'm not sure, maybe it has to do with the immune system. Have you spoken to your vet about this? You really need to ask for advice from your vet.
My cat used to have cat dandruff but he has never been overweight. I started giving him a cat bath once every couple of months with baby shampoo and that seemed to do the trick. It was difficult in the beginning but he's used to it now. I think he kind of enjoys a warm bath.