Turn the converter off and you will have two systems, 12v and 120v. The 120v will run the outlets when plugged into shore power, AC and all. A transfer switch could let you wire up the inverter so it would power the outlets unless you were on shore power.
I haven't gotten to that point yet. I plug the shore power cord into my big inverter while we dry camp. We turn that on and off as needed to run larger loads while the TV/sat/ phone is plugged into a small inverter that is always on. It can use less to run them than my last big inverter used just being on.
I suppose you could have the air conditioner wired so only it was on the shore cord but you would lose the ability to use shore power for the rest of the rig if you needed it. You would have to have a second box with a breaker to the AC and another to the power center to use them separately.