What does the other end of the drum-half look like? From what you said it sounds as if the outside of the ring gear is loose in the drum-half. If this is so, and it is not falling off the axle already, it is presumably tight on the axle shaft. It has a couple of ratchet boxes on that axle, which may be bolted to webs on the drum halves. That would explain why the ring gear is loose in the drum-half: its job is to drive the axle shaft, not the drum-half.
In the crummy picture in the parts list, it looks as if there is a web in the center of the ring gear, so the collar in the center of it has to come out toward your camera, not by pushing it down through the ring gear.
That suggests that if you try to push the axle shaft down through the ring gear, it will be difficult to support the far side of the ring gear well enough to keep from cracking it. In fact you may end up with a set of fractures of it, very similar to the set it currently has. In other words, maybe somebody hammered on the end of the axle in an attempt to remove it, without taking steps to support the ring gear from the other side. That probably isn't the way to go.
What prevents the axle shaft from moving upward in the picture (toward the camera, taking the collar in the center of the ring gear with it)? There are a couple of ratchet boxes on the axle shaft, driven by Woodruff keys, but it may slide through those by sliding the keys along keyways. So, can you support the inside of the ring gear by resting it on a ring slightly larger than the collar on the shaft, then push on the far end of the axle shaft?
