It kind of depends on what kind of wire feed you have. Some have a jog switch. If that's the case then I would wire in a remote for that and zap the wire to the dent and then bump the remote to feed the unenergized wire.
Without that feature things get a bit more complicated. There are two possibilities. One is to supply power directly to the drive motor. You need to determine what voltage the motor is and whether it's AC or DC. You'd need to find the source, bypass the controls and use your foot switch to feed it directly. The second it to disable the relay for the main transformer. One big enough to switch the secondary power would have contacts the size of silver dollars so things are done on the primary side. It should be easy enough to install a NC switch in the actuating coil circuit and kill the transformer when you just wanted feed. The problem is if the relay also controls any other parts. Go over the schematic and see if any of this makes sense.