There's only one solution at the moment, and it's not a cheap one unfortunately. You'll need to get a PWR-56 which has the repeater functionality built in. As you'd only be powering the PWR-56 itself and nothing else there should be no problem with powering it from a battery.
Someone here might possibly have one they could sell for cheap or (and you should definitely try this) try to get Ring to sell you a PWR-56 without the included PA-2 power supply as you don't need it. Explain to him you do outdoor G scale battery power and you simply need the repeater functionality as you do have a coverage issue. If he won't do it then ask him when he's going to make a pure repeater available then.
Now that said, I don't know how much extra coverage you'd actually get as I've never tested the range myself.
- Tim
