Re:RAT.
I have a suggestion that,surprisingly nobody else seems to have hit upon. Could RAT refer to 'THE NORWOOD BUILDER'?
This story is the second after THE EMPTY HOUSE in the Return of Sherlock Holmes series. Aside from it meeting the canonical
approach by Messrs Moffatt and Gattiss, the story's main villain literally spends most of the time Holed up behind a false wall in his
house,and when exposed at the Story's conclusion Holmes says to Lestrade and Watson  "Well, now, let us see where this rat has
been lurking."

I have read the other posts where it has been guessed that RAT may refer to 'The giant rat of Sumatra' or 'The Boscombe valley mystery'.
However,i believe the case for 'The Norwood Builder' is much stronger for the reasons given.
A: The story is canonical,directly after Holmes' return.
B: The fact that the Villain is Holed up like a Rat behind a wall.
C: That Phrase again by Holmes: "Well, now, let us see where this rat has been lurking."

I could be completely wrong,but am glad to come up with something different to dwell on.