There's another option that you might like for a couple of reasons: treadmill motor. I've replaced mill, drill, and lathe motors and have 2 bandsaws in the queue. I have a couple of 3 HP, and have seen a couple advertised, new, for under $250. If you're willing to look around and accept used, keep an eye out for out-of-shape people sauntering around in expensive jogging clothes. They thought that if they bought the super-delux model and wore the high-tech duds they would enjoy exercising every day, but now its just a clothes rack and they'll sell it cheap so they won't have to confront it every day. With the money they get, they can buy a new pair of cross-trainer shoes to keep up the image.
The DC motors will run both slower and faster than you need without changing a belt, have infinite speed control at the twist of a dial so you never run the wrong speed because you're lazy like me or in a hurry and don't want to take time to switch, and can tweak on the fly like when you're facing a large diameter and need to run slow on the outer but can speed up at you move in. The low end torque is amazing - you can wind some pretty stiff springs with it.
They run on 120 VAC at up around 20 A, run through a rectifier and motor controller. I got up and running fast using just a HF router speed control $15-20 and a full-wave rectifier $5 or so. That's the cheap and dirty way to get by until you decide how you want to do it in the long run. You can't go as slow or use the full power, and it may hum a bit because the DC is unfiltered, but I still use it that way on my drill press after a couple of years. Long term options are buy a commercial DC motor controller (Amazon, Alibaba, etc.), make one from circuits on the web, or adapt the one that came with the treadmill. Oh - and here's another plus: you can set up a relay that removes the power when you step on (or step off, if you prefer, as I do) and puts a near short across the motor. That acts like a hard brake. I can stop my 12x31 lathe in about 1-2 seconds (yes, that's not an error, one to two seconds, depending on the mass of the work). Mount a microswitch at either extreme of table travel and wire to the relay, and you'll never run the cross-slide into the chuck again. It sounds more complicated than it really is, the hardest part is fitting up the pulley and customizing a bracket so it lines up with the spindle pulley.
[added afterward: the dc motors are a lot smaller physically for the same HP - don't be alarmed. Also: they usually come with a heavy combo flywheel-fan-pulley that you will almost certainly remove and replace (trick - it's usually left-hand thread and comes off easily if you speed the motor up forward and then stop it fairly fast). But - that means no more fan. For light duty, you don't really need it, but for serious or prolonged work, attach an ordinary 12VDC 4" computer fan on the bottom of the motor and run it whenever you're at the lathe. That's a much better solution than a fan on the motor shaft because you get full cooling regardless of the motor speed - an important thing because you'll often do the heaviest work at slow speeds. Another trick: you can monitor the load on the motor with an ammeter in series with the motor - use a DC ammeter at the motor, or AC model ahead of the rectifier-controller. I use it when I want to max out the power just short of blowing the 20A fuse.]