How goes your project?
I'm interested in this as well. So far, I've collected the outline and things to consider.
First impression is capacity of a microwave transformer and gauge of metal you'll work with.
Next is what your electrical service is. Bigger handheld units are 220 single phase. I wouldn't want smaller.
It also seems like an arc welder with incremental amperage would be a good source of power, build a tool that plugs in.
Building one, the main task is the squeeze and contact mechanism.
There is a break in commercially made welders from fan to water cooled, related to power and duty cycle. Water cooling runs a little heat exchanger, a auto heater core size, and jacket is within the arms holding the copper tips.
At work, ours has a timer sensitive to partial seconds, with a vernier style knob. Newer machines are probably digital.