Newer 'power centers' auto switch, if you are plugged into to 120v AC, the converter is powered and supplies 12v DC to the circuits including to the battery.
When you are not plugged into 120v AC, the battery provides the 12v DC to the circuits.
You'll likely need a 30A cord wired into the AC side of the 'power center' to be able to plug into shore power. Your existing 120v AC circuits are wired to the breaker/buss bars.
The 12v DC circuits would be wired to the various circuits available, the battery has its own, they share a ground.
If you have a safety kill switch for the lights (kills power to light when not popped up), you'd wire the light circuit through the kill switch.
