Yes it had to be a SA gearbox, Nortons introduced the "Dolls Head" box circa 1934 to improve and enclose the gearchange action, and also get away from the hand change era.
This was because Sturmey Archer stopped all motorcycle manufacturing then, and this included no longer supplying gearboxes to Nortons and others. Nortons bought the SA design, improved it and had it built for them elsewhere - by Burmans as it happens. (although there is nothing Burman about the design.)
The long lever throw is about how they came.
As said, you'll get used to it !!
You may be able to do an outright swap for a later dolls head box to suit your era of bike, early 1930s boxes aren't that common, and folks are probably looking for them. Whether this would include cash either way is another matter, may depend on the condition of each box, and yours sounds to be pretty good. Ex-WD boxes are often extremely worn, so watch for this if you go down this path.