Billy,
If running a dedicated 240 volt drop is practical, I'd do that. Then you don't have to worry with any other considerations. You didn't say what HP your motor is but the Clausing 8520 manual says 1/2 or 3/4 HP is required so probably one of those. Whether you can run those on an existing 120 volt 15 amp (or 20 amp for that matter) circuit depends upon what else is already on the circuit. My shop, formerly know as the detached garage, has its own 60 amp feed and as I did all of the wiring in it, I set it up with dedicated circuits for lighting, wall outlets, AC/heat and fixed location equipment. But your situation may be different. If you decided to rewire the motor for 120 VAC, and run it on an existing 15A or 20A circuit, you would need to find out what else was on the circuit and consider whether any of it would be running at the same time as the mill.
Theoretically you could in fact run a 3 HP motor on a 120 volt 20 amp circuit. 3 HP mechanical converts to 18.65 amperes electrical @ 120 VAC. But that assumes 100% efficiency in the motor and capacitor start single phase motors are notoriously inefficient. A quick search on Grainger turned up only one 3 HP single phase. It's FLA is 33.8 for 57.8% efficiency. That is quite good efficiency for a single phase capacitor start motor. Several smaller motors I looked at are even lower. And it also depends upon how hard the load is to start. Fortunately, a mill can generally be considered easy starting as there isn't usually much mass to accelerate.
Robert D.