Dave, I did wonder if that was the case with the armoured cable.
One thing I wonder about, running a motor off an inverter, isn't that 1Kw is enough to run the lathe, but whether it's enough to handle the surge current when the motor starts up.
My Myford has a 570w motor, but if started conventionally, would probably trip out a 1Kw inverter.
I wonder about the possibility of a 240v soft starter, or using an inverter to power a single to 3 phase VFD, as either of these should help smooth out the start surge.
What I don't know, is how either device would run off an inverter; I guess you would need to consult the manufacturers of both the devices you proposed to use.
Incidentally, as I'm sure you are probably aware, the same issue might arise with a generator if it didn't have enough current headroom. My little 600w Honda suitcase won't drive a 450w angle grinder as the start-up surge trips the genny. It's perfectly happy with a 600w variable speed electric drill, even when drilling maximum sized holes, as effectively, the drill has its own built in soft start.
Bill