My two cents! If you weld the axle, let it cool very slowly, in fact, insulate it and let it cool down as slowly as you can. You might consider preheating and slow cooling like you would welding cast iron. Welding brings the steel to the point where there is a crystal structure change and quenching it will leave it in a hardened state. The axle does not have sufficient carbon to be truly heat treatable but it can be effected to a lesser extent. Along the edges of the axle it will cool too quickly if you don't find a way to slow it down and the edges will become harder. This harder surface allows the creation of small cracks which become stress risers. Don't let the axle change temperature rapidly, let the structure normalize. No formal education in metalurgy but a background in materials. If you buzz box weld it, be sure to use low hydrogen rods.
