Our band DPN+1 do it in C, which fits the voice and the rest of the group (I will give chords, if anyone wants - but easy to transpose).

I've always looked upon it as a "pop song" (and a very good one), and why not. It depends how you interpret it. Most people prefer the jazz/swing approach - Fats and Billie were famous for making decent jazz out of pop songs.
The version that didn't work was the version recorded by one Tommy Bruce, that was a UK hit in late fifties. A good pop song murdered....
(The chords sequence sort of fits "Making Whooppee" too, which is useful if you get a request for the latter.)

PS Why did Fats never record "Black & Blue" - or maybe he did ?