Russ. Looks like you have three threaded locking rings. You only need two, and from your description the assembly of the parts is not in the correct order. The bearings and threaded locking rings are the same on both sides. The locking rings have left hand threads.
Start by inserting the ring spacer 65-5884 into the splined end of the empty hub. This ring spacer is described as a hub bearing thrust washer. Its just like a piston ring with no gap! Drive this into the empty hub until it seats on the machined step. This locates the bearing outer race. Follow this with the bearing and add the threaded locking ring retainer. Tighten the retainer down to locate and clamp the bearing outer race against the ring.
Turn the hub over and insert the tubular spacer, 67-6061 ("Distance tube and retainers"), followed by the other bearing and then its threaded retaining ring. Avoid overtightening this second ring, too tight puts a sideways load on the ball race, just nip it up snugly, use Loctite on these threads. The tubular spacer should end up lightly clamped between the bearing inner races. That's it.
BSA Service Sheet 212C gives a cross section of the hub, but poor detail about re-assembly.
Plunger hub has two threaded retainers, one each side. S/A has only one, on the splined end. Here the other bearing outer race "floats" in the hub tube, has no threaded locking ring but is retained by the design of the hub spindle sleeve, 67-6027.
Bearings are the same for both hubs, but the tubular bearing spacer is different, so if you still have a problem could be you have a mix of mis-matched bits.
Swarfy