i am not sure. but try to implement with help of zener diode across output of power supply rectifier.

You probably can't do it, if it's a magnetic shunt design.  Most AC welders use a magnetic core which is not like a transformer.  The current control is via the magnetic field.   The only way to change this is to modify the magnetic core itself.    Why not just find a used utility transformer?  (I went the other way and made a welder from two old utility transformers.)

Interesting question. 

You may consider connect a shunt circuit at output of your current source. Use a voltage sensor at output and feedback the signal to control your shunt. Hence the shunt will shunt excessive current to keep the output voltage constant. 

Alternatively, forget about your current source, buy a DC power supply, and make your life much easier.

Crazy idea. I feel difficult to understand the concept of welding with a constant voltage source.

The weld resistance is more before the weld starts. once the melting of the parts to be welded starts, the resistance of the weld decreases rapidly. The constant current sources are designed to supply constant current even when the weld resistance falls, thereby controlling the heat of the weld.

Now when you use a constant voltage source, the current increases during the melting stage and the control of heat becomes impossible, and the source may burn out due to excessive current. you need an external agent to control the weld current. 

If you are trying for another logic to control the weld current, then it is OK. But I doubt the need.....