Well, the same names do seem to pop up in Watson's narratives - here we have Trelawney Hope and there was a Hope in STUD. So it might be possible that he gave all the detectives pseudonyms to protect their identities, and here he was simply using the same pseudonym twice.

I must admit I hadn't considered that Bellinger might discreetly remove Hope from office - the story does seem to indicate the Prime Minister has faith in Hope despite what's happened. But yes, it does appear to be a possibility - Bellinger may suspect a member of the household must have been involved, and so doesn't want to take any further risks - wants to tie up a loose end.

...every beauty of body and of mind... Watson does lay it on pretty thick, doesn't he? ^_^ But a different time I suppose - men were able to show appreciation of another man's appearance without it seeming automatically sexual.