AC is AC but if from a mains suply one will be referenced to earth, if through a transformer there is no earth reference ( unless someone has earthed it), but then you would need fuses/protection in both circuits.
I commented on a comment made about a video. The video referred to the situation in the US.
not sure how the 50/60Hz difference will affect the weld.
It won't affect the weld and it may not have any affect at all. Most welders are specified for 50 or 60 Hz, this one isn't and it is a cheap 120V Lincoln. If the transformer is made cheaply, its impedance at 50Hz may be low enough to cause heating. It's probably just me, I don't use climbing equipment that is not rated either.
Jack