The characteristics of the welder itself will not be changed.
The wire size will remain the same with the exception of the equipment grounding conductor, which with a 60A overcurrent protection could, by NEC, be reduced to a #10.
You've just had an easy hundred or two years of combined experience explain this to you.
Are you sure you want to put that 40Amp breaker in the subpanel? You've now gone from a matter of the manufacturer's recommendations at 80A OCPD with #8 to a 40A OCPD with #6??
Does your foreman have any reason that he can cite for the increased conductor size to #6? That's just nonsense.
