1968 Powermatic model 90 conversion

[Bill Howatt]

I'm going to amend my original post, while VFDs are not recommended for single-phase motors, that is for the typical variable-speed function which is why most turners want them. If you are just going to run at the single 60 hz frequency and are looking to boost the voltage then such a drive can be considered that provides the capability. However, it is not just a matter of getting 240V, you have to consider the power the motor will want, so at a minimum, the current input to the VFD will be double the normal current. This could be a concern regarding the size of 120V wire and breaker feeding it plus what other loads are on the existing circuit(s).
In the title you did specify the lathe make which I assume is the standard configuration of a motor with any speed alteration done by pulleys or Reeves mechanical drive, not electronic. This would point to perhaps the generator is not up to it.
Obviously, you are going to need further consultation with an electrician.