I've only used the word "gentle" as a verb in that sense, to gentle a dog; never used "gentling" as a word to describe the person who does it.

I'll exclude the "pet psychic" as well as "ghost whisperer" and guess that those are much better included under the umbrage of "charlatan."

(There was a tragicomic episode of dog whisperer where Cesar is helping a professional "pet psychic" with her own unmanageable dog.)

Best guesses: Trainer, empathic trainer, empath, animal behaviorist, animal psychologist,