Years ago I considered purchasing a mini-lathe and talked to several friends who had one. They all said the same thing - whatever size lathe you buy, you'll always find a project that's just a hair too big to fit.
I took those comments seriously, and instead I bought a used, 1959 Clausing 12"x24" Lathe.
The unit came with a 5 HP 3-phase motor in the unit but as part of the sale came a 120V240V 5HP Baldor single phase motor. A little rewiring and a few bolts and I was in business - I used the 3-phase fwd/rev switch to achieve the same thing with the single phase motor.
I am very glad I purchased a older, larger unit for this machine has become a workhorse with my metalworking. I've made spindles, axles, hubs, wheels, pulleys, tensioners, mounting plates, pistons, bored the cylinder in small lawnmower engine blocks, made valves, rollers. I've even turned acme-style threaded rod out of round stock with decent success for converting a friend's plasma cutter to a CNC cutting machine. I've turned many, many brake rotors and drums over the years for various cars I own, even though sending it out for $10 was entirely doable and far less work. It's a fun machine to use, and I enjoy using mine.
I'm really glad I bought the 1959 Clausing - cast iron everything, rigid as heck, and the one I got was barely used - the prior owner bought it new and passed away in the mid 60's, and it sat in his son's garage for about 35 years before I bought it from him.
Phase converters do work however you have to size them appropriately - not too large, not too small, otherwise you'll get little power and make a lot of heat. I would convert the machine over to single phase with a new motor. You can buy quality single phase motors on ebay typically less than a motor supply house. Another option is to find a motor/alternator repair facility and see if they have any rebuilt motors - you'll save about 40-50% that way off the cost of a new, equally sized motor.