I still think the root cause of the problem is the 40 Ft extension cord. With the start-surge the voltage drop could be considerable. This in turn could cause the start surge to last longer.... popping the circuit protection.
Could be, but I think the problem is the inverter, not the extension cord.
On a mountain property I unrolled 250' of 10 AWG UF, from an electrical outlet to a South Bend set up inside a cargo container. Actually, to a trailer first, then extension cord to the lathe.
Consider a 100' round-trip 50 ft, 16 gauge extension cord:
AWG Chart of Approximate equivalent Cross-Sections of Wire, American Wire Gauge by Resistance per 1000 feet
www.interfacebus.com
That would be 0.4 ohms. Try to put 50A starting current through it - that's 20V drop, so 120V in gets 100V out. Probably enough to start.
Sunburnt said the inverter was in another room. Probably going through 12 AWG house wiring, not an extension cord. That's 0.16 ohms and 8V drop.
But sure, thicker gauge and shorter is better. I have many 12 AWG cords besides the UF I sometimes use for semi-fixed applications.
The inverter on the other hand probably can't supply 6 kW for more than a few milliseconds and voltage collapses. Induction motors are the problem; brush-type would work better. I had thought a VFD and 3-phase motor would be the way to go but its switching upset one of my inverters (maybe an inductive filter on the input would fix that.)