J in the UK
I second Dugaboy's opinion of Champlin Arms, fine people to do business with.
For what you have stated any double in calibres from 280 to the 450/400 would do a good job for you. Especially the 400/360, 400/350, 360 No2, 333Jeffery Flanged, 375 Flanged, or a 400/360 that has been [properly] rechambered to 9,3x74R. Even a 375 Flanged Magnum could be loaded with 235 gr. bullets for the smaller deer.
I was unable to fine a good medium bore British Double rifle so I bought a Chapuis in 9,3x74R. It has become my most favorite rifle under 40 calibre. It ain't British.... but it is a very fine hunting rifle.
Keep up the Quest.... And let us know what you eventually get.