The details you give does not sound like a PID loop can control it.

I'm not an expert in welding machines, but I do also have some doubts...
Assume for instance you have a short voltage dip on mains. If now the PID loop shows some relevant differential term, then a much higher power will be applied on next spot welding event. But what if then mains voltage dip has dissapeared? What if there's suddenly a voltage spike? Then, welding spot would be extremely overrated and burn down.

I think a better approach is to design a constant current source, means to control firing angle every half period in order to keep welding current through welding spot constantly, and this for the duration of actual spot welding event.

Kai