I make no claim that anything I am about to say is code compliant.
A rating of 600V 60A generally means that the switch will interupt that current at that voltage without continuing to arc and without allowing any leakage to ground or across the line. Most switches will carry higher currents than they will interrupt. At lower voltages the arc when the switch opens is easier to extinguish so lowering the voltage should allow some uprating of the switch. Careful reading of the label may indicate this. If the disconnect has fuses in it you should be sure that you can get 125 amp fuses in that size. One suspects that that is a bit more than the expected draw with the machine set at max and the electrode shorted to the work. Most likely the disconnect would not be operated while the machine is welding so one would expect the draw to be the idle current, probably a few amps, and easy for the disconnect to handle.
Now the dark side- if that building burns down for any reason, and the insurance investigator finds that that switch is underrated you may well wish that you had gone the extra bucks and gotten something bigger.
Bill