As others have indicated, a VFD is the best solution. And yes you will need a new motor. I've gotten them off ebay for a very affordable cost.
I can only speak from my experience on a 10X24 bench lathe, I've owned since new in 1976. It had a single phase motor, and it was not the most difficult to change the speed of the final 3 step V-belt drive line. But from the motor to the jackshaft, it had an addition 2 V-belt selection. That did require messing with the motor mount bolts that tensioned this initial drive belt.
So about 25 years ago, when VFD controllers were well into being commodity items, and affordable by the home shop, I purchased a 1HP 3 phase motor and a VFD from a US company. It was a package special (dealerselectric.com). This VFD was not of the 'vector' type, and so low speed use had a cogging effect. But it worked great outside of trying to use it below 20Hz operation.
Since then, I've replaced every single phase motor in the shop with a 3 phase and VFD of the vector type. And as the motors are 4 pole, the phase speed for 60Hz input is ~1720RPM. I program the drive to allow 120Hz max operation, this allows the motor to run at ~3600RPM, giving a very wide speed range. I typically set my V-belt reduction to the mid range of the machine.
When you know this reduction ratio, you can program that into the VFD so the LED display will show spindle RPM, which is very handy on the mill and drill press for setting the correct SF/M (surface feet per minute, or metric equivalent in your country) for the cutter diameter in the machine.
My milling machines (Wells-Index 645, and a 1930's Rockford Mill, horizontal) came with 3 phase, and I initially used a home built phase converter to power (I would never go back to a RPC, no speed control, another noise and power burn in the shop).
I also have a lathe with a 7.5HP 3 phase motor, that is powered with a VFD of 10HP capacity. Typically once you get over 3HP, VFDs want 3 phase input. And the derating is typically 2:1, assuming you will want to use the machine at its full capacity, when feeding them single phase input, as it is the input rectifiers of the VFD that can be damaged. So my example is not the 2:1, but I've never had a shut down overload of the drive, and I've done some pretty good depth of cuts. VFDs can be oversized, and you program the parameters of your motor, and it will prevent overload damage. At very low speeds, motors may overheat, but I'm not doing production, or running the machines hard in my shop, so I've never seen a hot motor from low speed lack of cooling.
The best solution is not to mess with a 120V input VFD, and just accept you need a 240V power jack at the machine. The 120V input VFDs have a voltage doubler, as they are intended to drive a 240V 3 phase motor. It works at low HP levels. And I see if this is the path you want, here is a link to an affordable solution (currently out of stock) https://dealerselectric.com/1-HP-1800-RPM-115-Volts-Input-Package-4.asp this includes the motor and VFD of a quality manufacture.
Installing a VFD will require rewiring the machine, I could not find a wiring diagram for your lathe on line. The owner manual and parts list did not have this, so I don't know how difficult integrating a new motor to your existing power will be. You do need to look at your current motor nameplate, and find the frame number, as well measure the shaft diameter, as this will make putting a new motor in its place easier, as having it bolt in place of the old motor, and reuse the V-belt sheave is the desired path. You want to measure the mounting holes and locations relative to the shaft, and then look up an equivalent in a NEMA chart, to ensure a new motor fits without excess effort. At worst you may have to make an adapter plate, and purchase a new sheave.
That's my input to this post.
