What will happen in theory if a 240V single-phase AC motor is run on 120V?
Testing the motor on your workbench at 120 volts should be as safe as your test setup. Ground the motor frame. Fasten it down so it doesn't roll off the bench when it starts. Be ready to switch it off if something goes wrong. The motor should produce about 1/4 of rated torque with 1/2 of rated voltage. If the bearings are ok and the switch is ok, it should accelerate to near rated speed and activate the centrifugal switch. Be ready to shut it off if the switch doesn't activate right away. You should monitor the current with an ammeter and be ready to shut the motor off if it doesn't drop to a low value quickly when starting. The starting inrush will be something like six times rated current.

If that all goes ok, you should monitor the current and be ready to shut it off the first time you try it at 240 volts.