I think I've rebuilt 5-6 BA/CP gearboxes and I cannot remember any of them having the steel sleeve/bush tightened hard up between the inner race and the splined racket gear as Ian suggests. It is possible all my bits were worn out...but I didn't think that at the time, so this morning I've taken a closer look on the bench with some spare parts I have:
The first photo shows the protruding mainstaft in the outer space with the steel sleeve/bush and splined ratchet gear in place and the nut tightened down. For clarity, I've left off the spring and the rotating ratchet gear.
On the other side of the case / mainshaft, I've fitted the correct un-splined gear:
With this current set up, I can easily detect endfloat...I can push the mainshaft in-and-out of the case cover and I can measure it as 21 thou of an inch:
But should there be any? Absolutely! If I simulate removing all endfloat just with my fingers, that un-splinned gear on the other side of the case cover soon locks up and is no longer free to rotate on the mainshaft.
I don't think 21 thou is excessive but it is probably more than Burnam intended as my steel sleeve does show some signs of wear, as does the underside of the splined ratchet gear and the face of the inner race...plus the two faces of the unsplined gear on the other side of the mainshaft.