1968 Powermatic model 90 conversion

New to Sawmill Creek. Not electrically gifted. My wonderful 1.5 hp lathe was running on 240V single phase plugged into dryer outlet via 10 AWG extension cord successfully for 2 years for our wood baseball bat company. Moved to new outbuilding where nearest 240V outlet is 100 feet away. I have a junction box in new outbuilding with 2 x 120V via conduit running under driveway. Was going to run a 10 AWG 100 foot extension cord to dryer outlet in house until I got an electrician out here but heard voltage drop might create a hazard. Bought a 4750W generator (because always nice to have a generator) as another stop gap and power flows and lathe spins but contacts in lathe flutter and don’t stay closed and generator seems distressed and hops around. I used three wire 10AWG in extension cord with an L14-30 plug for the generator. I have a 120V outlet 6 feet from the lathe. Wondering if a variable frequency drive is in my immediate future. Thanks in advance for any help you might be able to offer!

Variable-frequency drives and single-phase motors don't mix -they are used with 3-phase motors. There are some that say they operate with single phase but they are expensive and can be very selective about the motor. I wouldn't pursue this. So what is the size of the cable and breaker feeding the 120V line?
You say there are 2x120V, if they are not on the same phase you will have 240V from line-to-line and so there would be 240V available. A 1.5HP motor will run off a 15A 240V circuit as you know from the previous setup.
The generator must not be putting out stable or very clean power if it causes the contacts (I assume there might be a magnetic switch but I would think it would just drop out and stay off) to open and close which would switch the load on and off the generator causing it to jump. If your lathe has electronic controls then the dirty power from the generator could cause the circuit to go nuts and even possibly damage it.
What make and model lathe is it?
First thing I'd do is check to see if the 2 120V circuits will give you 240V, and if so, the electrician should be able to get it setup. However, I'm a bit surprised he didn't mention that so maybe there aren't 2 120V circuits that will give 240V. Note, if the problem is that there are 2 but on the same phase he should be able to move one to another phase.
I am not an electrician but a calculator indicates that for AWG 10, 100 foot run, 15A load single-phase the drop is under 4V which isn't much although we have to allow for a starting surge. I have a 3/4 HP, deep well pump that has a run of about 400 feet on #10 wire. Another comparison, my 2 HP lathe has a run of over 40 feet from the panel on only #14 wire.

Not all generators are created equal. I bought an inverter generator for the cleaner power situation for computers in the house. My feeling is that there are good people here, but not many professional electricians.

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.

I think a simple solution for single phase may be a step up transformer. A 1.5HP motor is 1,125 watts. I'd add a little safety factor to that, say 1,500 watts. You can even buy these units with a 120 pig tail to plug into your outlet, with a 220 socket for the output. You'll need to make sure your 120 supply can deliver the amperage you need, which for 1,500 watts at 120v is 12.5 amps. Starting a single phase motor with simple contacts may exceed that capacity due to inrush current.

Thank you for your input, gentlemen! Bill to your question, the wire feeding the 120V outlet next to the lathe is 14 gauge, white Romex. The subpanel in the barn is a 15A Square D which runs to a 15A double throw in the house main panel. Excited we were finally about to get an electrician out here since we last spoke who was willing to work with me in the rain to remove grey UF and a line of white Romex (which I understand is not rated for burial) from underground conduit. We replaced that with four strands of 6AWG from the house panel, through the conduit, and connected it to the barn’s subpanel. Today, while watching YouTube I came to understand that “High Voltage” and “Low Voltage” wiring diagrams printed on the motor housing refer to, in my lathe’s case, 240V and 120V, respectively. I rewired the motor for 120V, spliced the motor’s power cord to a 16AWG 120V extension cord, grounded both to the lathe, and plugged it into the 120V outlet of my new Wen 4750W dual fuel generator. Same distress of the lathe motor and hopping around of the generator as pulling 240V from the generator, regardless of powering with liquid propane or gasoline. The motor pulls 8A at 240V and 16A at 120V. Very frustrated, I plugged the lathe motor into the 120V outlet on the wall and the damned thing quietly jumped back to life. Obviously the start and stop switches do not work but the Reeves pulleys worked as they always had to adjust the RPMs. I expected the amperage to trip the 15A breaker or the Romex to heat up, but neither happened and I worked on an 11” diameter log, turning it down to 3.5” diameter billet! I stop the machine with the power switch on a power strip. Far from ideal but a stopgap until the electrician comes back out on Tuesday to finish installing a 240V outlet next to the lathe.