It's quite unusual to find a single phase VFD above 4kW or so - and usually of course single phase tends to mean 240V. Looking at the screenshots, this Chinesium example doesn't seem to check for loss of one of the incoming phases, so as long as it sees enough voltage on its internal bus, it will presumably run. I guess the question is whether the undervoltage limit is set too low for it to run off 240V. The other limitation is the maximum phase current on the output. It may be limited to a level that would restrict the net power when running with low input.

A rough and ready solution might be to boost the input from 240V to 360V single phase by connecting a 3kVA 240-120V site transformer as an autotransformer. - this would allow you ~9kW capability and require perhaps 40A total from 240V, if you ever went there.

But when do you really expect to be running 6kW at the spindle? This would mean an heroic material removal rate by our standards. I have a 300A TIG welder that runs off 240V single phase, which requires something like 70A at full chat - not a problem, as I have a 63A feed from my workshop consumer unit - it's perfectly possible.