What kind of specialist diagnoses channelopathies? Neurologist? Rheumatologist? Both? Some other specialist?

Basic research. There are many people working on channelopathies . It is a way to separrate problems by what is though is the mechanism of action of their disease . Depending on the diagnosis , the specialist of that disease and many time the pediatrician is the one that sees this problems.
Some advice. The diagnosis of a channelopathy is the outcome of testing and evaluation, but the process starts with the organ system displaying symptoms. So if you joints hurt, see a rheumatologist, if you are numb and weak, a neurologist.