Stick to JS's advice and get a 1hp (4 pole 1450RPM) motor, it's plentiful for the work that's done on a 3-5" centre lathe.
The motor overheating sounds like it might be a problem with that particular one, but not a reason to up the power as such, if you bought a new motor a 1hp should be good enough.
I have a 3 phase with a VFD and i'm very happy with it, the speed control isn't a catch all as such, on large iron castings you still have to change the belt to a happier setting for that speed.
I have no particular hatred of single phase motors though, I just find 3 phase lighter, safer and nicer for operations like screw cutting and tapping and reversing.
Michael W
Edited By Michael-w on 25/07/2017 15:11:55