I can’t see anything in the VIRM that says they need to be independent to the headlights. Modern motorcycles often have a park light and headlight on at the same time with no option to have just the park light unless the engine is off. But I would do as Igor suggested.
https://vehicleinspection.nzta.govt.nz/virms/in-service-wof-and-cof/general/lighting/forward-facing-position-lamps
My bet would be a bad earth between a bulb cap and the bulb holder. When you ran the test earth cable from the battery are you connecting it to the bulb?
I have one of those, but haven’t used it as much as I should’ve. Got it from these guys as it was the best price at the time.
https://www.prolineindustrial.co.nz/equipment/welding-equipment/mig-welding/mig-welders?viewmode=grid&orderby=10&pagesize=20