I have several times tried to piece together this Sutton information, which you have nicely delineated. I also agree with JF that the names are likely Sutcliff and Clifton. But I'm puzzled by your conclusion here:
"I conclude the other is a psychiatrist, called in to help Hazel and never mentioned by Shade."
Kinbote says in 230 that because the Shade's disliked "modern voodoo-psychiatry" they "could not do much about" the situation. This suggests to me that John & Sybil did not consult a psychiatrist at all; therefore, psychiatry could not have been Dr. Sutton's field. Or are you saying that because Sutton was "old-fashioned," he may not have been a "modern, voodoo" psychiatrist but instead a more reliable sort?