Open to any ideas. Only have a mig though.
Well, especially if you've already gotten one side out, then you have a good sense of what is preventing you from pulling the other side out: that weld is acting as a fat ring that's preventing that axle from sliding out of there. By whatever means you have to get that weld cut down so that it's no longer has the ring and the axle shaft joined together, and it's no longer too fat to slide out through the hole in that gear.
My best guess at this point is the long neck die grinder like I described above. They make them in both air and electric, either one would work.
As far as the disassembly, other people would know better than me, but I'm pretty sure you can completely disassemble that rear end in place without removing the backhoe. Once you've pulled the axles out all the other gears and the carrier I think will come straight out the top. And you obviously already have the top open.
It does really look like this thing has had some water sit in it, so your gears and shafts are probably not in good condition. So I think you'd be well served to inspect all the contents and put it back together with parts that are up to spec. Whether that be new or used.
If I were in your shoes, I think I would reuse those existing axles and just weld it back together when I'm done. It's way less than ideal, but finding the proper axle for the backhoe model can be really hard to do. If you could just spend a couple hundred bucks and get them off the shelf that'd be great but you can't. So the welding may have to be good enough.
The one caveat I would make is that I don't think I would weld it together until I had function tested it with the wheels up in the air. In order to be absolutely certain that I have it back together correctly, I'd want to power up the machine and drive the wheels forward and reverse in both gears before I actually put the final weld in place to lock the axles to those bevel gears.
Of course if I personally had the machine here, I would weld build up the axles and machine them back to factory specs and reinstall them with the proper c clip. If you know a competent machine/job shop in your area, they could do the same. I'm not saying it would be cheap. But it is a bird in hand, and the job would then be done correctly. Especially if you're going to run this machine enough that you'll need to redo those bearings and seals at a later date.
Bob