I would not trust welding the tube to the housing, The housing is cast iron and the tubes are steel, they will not properly weld together. Find a axle at a junk yard, the earlier axles had better quality. The earlier ones were made in Mexico and the newer ones in the US. With production costs higher in the US they run the feed rate on the boring machine faster resulting in a less sound press fit
Its been done many times....considerable debate about what method is proper...Here is what I did:
Today Pop and I worked on the axletubes.
We pulled the axle out of the frame, and got it on the bench..yep same hi-tech bench as used for welding

We burnt the paint off the tubes and center section to be welded, then preheated the center section to ~600* right at the gap between the two pieces. Then I welded with .030 wire and highest heat, and about 7 on the wire feed (1-10). The first pass went into the groove of the tube and center section.
The second pass I tried to tie the center section to the first pass....had the wire feed up to 8, and it was too much feed I think, so we went back to 7.
The last pass was to tie in the first pass with the tube more. Wire feed left at 7.
The other side I went first pass in the gap, 2nd pass on the tube, and 3rd pass on the center section. The results were a little better.
So we did one side at a time, until the temp was back down to 100*, and the tube and the center section were never more than 20* different. We hit them a couple times while cooling down with the torch to keep them from cooling to fast.
We checked the temps frequently with an IR gun.
The cleaned up finished results:
Acetylene torch for heatup, and a couple maintenance hits to help it cool a little slower than normal.