I use an "inverter" that works on the voltage-doubler principle - it was a bit of a bodge by a now-out-of-business company who grafted a voltage-doubler to the front of what is quite a nice ABB 380V VFD but it does mean single-phase 240V in and 400V or so 3-phase output. It works fine until you put a heavy load on it when the effectively half-mains frequency input means that the DC voltage sags too much between cycles and the VFD trips. I've been kicking around the idea of finding a 3KW 240-415V transformer and go back to the original ABB VFD input arrangement - and a cheap transformer has not found its way to my workshop yet - or try one of those Ecogoo 240-380V boxes (I linked to this company in an earlier post on this thread). They seem to deliver a lot of performance for the price, if they work as well as they claim...

My lathe has a mechanical variable-speed mechanism so I have no experience with dual-speed motors and VFDs. In fact, I run the VFD at 50Hz output, with a switch that drops this to 25Hz for a quick low-speed option.