Somewhere all the wires get tied together... And, I find it amazing that the power companies don't seem to have to use the same gauge of wire that the code requires for mastheads.
But, your grid-attached system should allow you to pull adequate power from the grid.
How thick of steel are you welding? Amp draw?
I've welded from batteries using my cheap Chinese Modified Sine Inverter Plus Indian Transformer.
It seems to do just fine, although I get a bit of a voltage drop at the batteries, and I seem to max out a little higher than 100 amps.