MOSFET Gate Driver Selection
I would consider using an SCR on the discharge side for this type of circuit instead of a FET - particularly if you are not trying to do precise dwell times. The SCR is capable of handling very high di/dt, is easily triggered, and is self extinguishing when the cap voltage drops..

You mentioned that you are using supercaps as the welding current source. Keep the internal resistance in mind as this limits the maximum discharge current due to self heating. You will find low ESR caps but you must be selective. The datasheet will list the maximum discharge current.