It's the duty cycle of the machine that is the problem. They use aluminum windings. The resistance starts going up as the heat goes up. The arc current starts dropping off rapidly after a specific point.
The heat dissipation in the winding is a result of it's resistance. The resistance is a result of the temperature of the winding.
As the winding heats up, it's resistance goes up, which causes more heat to be dumped in it. That causes it to heat up faster. On and on.
There is a tipping point where it just goes into runaway heating. The only way for it to recover is to let it cool down.
The welds will all go to crap because the weld current will be a fraction of what it would be cold.
A cooling fan, will help.
Rewinding the core with copper wire will also help.
I don't think there is any way in heck you would run 1/8 on a big box store buzz box. They normally can't even run 5/32