The transformer would allow you to generate 120V from the 240 and avoid using a ground for a neutral. Still seems like a lot of effort/expense when there are other options. Converting the opener to 120V seems like a good plan if possible.
If you really are set on the transformer plan it might be better to convert the circuit to 120V and use a transformer to boost that to 240 for the door motor. You could use a 120:120V transformer and connect one side of the output to the new neutral. If phased correctly you will have 240V between the other output and the input hot wire. And the transformer only handles half the total power.
Several people have asked why not just convert the circuit to a 120V and use it that way. I did try to think about this. The garage is 40' by 30'. The garage height is about 14'. The door is a steel folding door that weighs about 2,500 lbs. Hence, the reason the motor runs on 240V. I don't think I have the option, at the motor, of switching it to 120V as it might not be enough power to open the door nor is this an option on the motor.
I'm not the original owner. But, I can't understand why the electrician didn't pull a cable like 10/3 originally, then just leave the neutral "unconnected". Maybe it was cost? I just don't know.
I do know that using ground for a neutral is not a smart thing to do, even if it will work. That's why I'm looking at the transformer method.
Maybe I'm just overthinking this and should just accept this.
