I think the wood in there would be a great improvement; and no, it probably couldn't be welded with a wooden insert glassed or JB Welded in.

If you would be satisfied with the top disc being brazed on and the bottom disc with a bead of JB Weld around the intersection this would probably be the easiest way to go with the most strength.
Yes, by all means remove the rear disc as well if you would be better satisfied with all welded joints. All of the brass would have to be removed in order to properly weld the joints with steel rod.
It would mean a bit of work but would solve all of the problems of getting everything lined up in order to re-install the handle on the blade proper...

Make an aluminum or steel tang replica the same size, length, and thickness of the original tang, actually make it just a tad longer so you can also get the step between the tang and the blade proper as it is on the blade itself.
Then reassemble the properly cleaned handle onto the replica tang as tightly as possible having all of the pieces aligned as close as possible and have it welded. After it is welded you could glue or JB Weld the piece of Hickory inside, drill it with several sized bits not going any larger than the tang is thick. Then use a wood chisel to remove whats needed to get the tang to fit properly on the original tang and then use the glass or the JB Weld.