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Being an old feeder it should run on 115 volts. So the first question is does you inverter have 115 volts in its feeder plug? Then you have to find out how the contactor circuits function. will a simple switch closure turn the welder on or do you have to return 115 or 24 volts, or maybe 42.volts or is it AC or DC.
if the SMW 35 is an older version of the mig35 it should use 115 volts AC with a 115 volt return for contactor. do a google search for esab mig 35. the manual for this feeder should be one of the first items found. look at address. should be ESAB.
One way to run the feeder would be get 115 volts from a wall plug use it to power the feeder and a relay. The relay gives you isolation. feeder drives the relay and relay contacts close and turn welder on.
Will need to know what welder model you have to be any better help.
Let me know if you feeder matches up to the MIG 35. If not I will have to go into the archives and see if I can find your feeder.