The "battery eliminator" is the USB plug. You can keep it charged through a cigarette lighter plug. When the battery is fully charged, the radio quits charging, whether the radio is on or not. If you turn off the ignition, the radio will run off the battery but will begin charging when you start the car again.
I also equip all my cars with cigarette lighter plugs that are always "hot," in addition to the existing plugs. I use fuse taps (available on eBay and Amazon) and aftermarket cigarette lighter plugs that mount under the dash.