Ahh, I read too fast, thinking the rear end in the bottom pic was from the car. I see now, so you already have the replacement. Good to know.

Welding the axle shafts together will give good traction indeed, and will lock the torque distribution to 50/50 between the rear wheels.

Stress on the mechanics might not be a issue if it’s heavy-duty enough. But tire/tyre wear and scuffing will be a issue on pavement (if its ever driven on pavement for a long time).

Sounds like fun either way.