Capacitors won't help you in a alternating current system in the way you are wanting.

You need a larger transformer and related wiring to handle the extra power demand.

No, it does not do, what you claim: the capacitor may act as a part of a filter, but it does not affect the sag.

You may be mixing apples with oranges. There IS a transformer, where a capacitor play a main role. Its magnetic core is M and I. One leg of the M is input, the other is output, the third is the place for a resonant circuit. That modulates the magnetic flux in a way, as to stabilize the output's voltage. Called resonant transformer.

Welding equipment has limited magnetic flux magnetics in them. Otherwise a short in welding would transform into a near short on the primary. Now, that would be bad form!

You also forgot to tell us, how much sag is allowed to these normally very tolerant equipment. You do not intend to mix other, less tolerant equipment on the same transformer, do you now?!?