Studies by the Electric Power Reliability Institue document that installing a ground rod at a machine tool (or welder) actually create more problems, due to voltage gradients in the earth, this ground rod adds to the noise the machine sees.
As many 250.54 electrodes (supplementary,I call them time and materials electrodes) can be installed as you want. There is little evidence that they are effective. Just don't install a ground rod and isolate the machine tool from its supply equipment grounding conductor. Thats a code violation as there is no ground fault return path.
Have you considered a UPS on the control circuit? It could even be a problem with EMI radiated thru the air from welding. Seems like the welding machine manufacturer could assist you.But if they say install a ground rod, go to plan C