Back to the main question though -
If you want continuous rotational positional control of your lathe spindle then you will need a sensor on the spindle and a motor/controller capable of the speed for conventional turning and also the hold for stationary and torque for slow mill-turn operations with live tooling, (that eliminates steppers) a controller capable of operating a brake on the spindle may be required.
Given the power in a motor required for slow moves and stationary hold for live tooling, even with a brake you will need a higher power servo motor than you think (multiple rather than fractional horse), I think your proposed budget may require expansion, you'll only really find out when you start asking for quotes.
Before you buy anything you should start with your chosen CAM software and proposed controller as this will have to deal with running the hardware and if your project ends up needing a 4/5 axis CAM and a genuine Fanuc controller you may find your budget not going far.