Depends on which switch you're talking about, and how your GD works.
My GD has two controls: a battery powered remote that I keep in the car and a button on the wall beside the door to the house.
The remote in the car has to talk to a receiver in the GD opener. That's not going to be on if the inverter is in sleep mode.
I assume this is the remote you're talking about, but in the same breath, the other button on the wall isn't going to work either, for the same reason.
I would try thinking about devices like the "clapper". Unfortunately the clapper needs the AC already to be on, but the trick is that a device like it, that you could turn on remotely somehow, is able to switch on a big AC electrical load, knocking the inverter out of sleep mode. So what you really need is the clapper that switches on an AC load (like a lamp) but is controlled by DC.
That exists.
It's called an Arduino (30 USD)
https://solarbotics.com/product/50450/Add a RF receiver shield (7 USD)
https://www.dfrobot.com/product-1089.htmlI just re-read your OP and see you don't have much experience with electronics. I suppose I could say this would be a great way to learn - the Arduino's are popular partly because they're easy to use, as far as electronic gadgets go.
Another way to skin the cat is to set up a "tripwire" in front of the garage door, basically using a second pair of the same infrared sensors that are used on the garage door already. These sensors work on 12vDC (mine do, so maybe you have the same type). Set them up so that the circuit is broken when your car is close enough. That controls a relay, and that relay switches on that AC light that wakes up the inverter and now the GD responds to your remote.