After looking at the comments of the ATO video, the motor can not be switched into reverse from the VFD. All it can do is control the motor rpm above the speed of the sprung switch for disconnecting the capacitors, even though they are not connected. To run in reverse the wiring of the motor will need to be altered.
So it looks like a single phase to single phase inverter can be used on a motor that has a cap start still enabled or having the cap start disabled. If the cap start is disabled, then instead of the 0.75kw unit, you will need their 1.5 kw unit. As they say that a 2 times the capacity of the plate rated power is required for the start of a singlephase no capacitor start motor.
On my mill drill recently, I did find a use to be inverting my saw blade and to run my motor in reverse. So for me, I will need a replacemant 3phase motor and single to 3 phase VFD unit .I am thinking of getting a 1.5kw motor that can run at the higher 2880 rpm. That way I can double the rpm and still have the same torque as the slower 0.75 kw motor at 1/2 motor RPM. (25hz the motor would be 1440 RPM) . The gear box lowest speed is still slow enough for everything I need, but the top end is really on the slow side, so that is why I am considering the bigger faster motor. They used to may the Mill I have with the 2880 2 speed motor but stopped doing that line of Mill drills. Maybe that could be in another complete thread. With the 3phase motor and controlled braking, I will be able to more easily do tapping of holes as well.
As for buying the VFD, I have bought all of mine off the local motor sellers. They used to sell complete VFD with a motor package, but no do this service. None of the vfd have ever given any trouble. They also sell the 3phase mototrs to suite the application etc.
ATO do make a single phase to 3 phase motor invertor and on their youtube video, do answer a lot of the comments questions about the setting of their drive etc.
Neil