I'd be a little surprised if there is a built-in
Inverter dedicated to an air-con unit, given how much power they draw. But it is possible if you have a large enough
battery bank.
If you want to have 240V sockets 'live' when not plugged in, then I would think a second
inverter might be the better option then messing with the existing one as it is possible that is a customised
inverter designed for AC use to deal with start-up surges and the like.
The point about having an
inverter feeding the 240V outlets which then feed a
battery charger and you end up going in a loop is an interesting one and is possible depending how the system is setup but best avoided as a possibility really.
The approach I prefer over messing around with cables when I want to use the
inverter is to use a Contactor, which is essentially just like a relay, that feed the 240V sockets. When the Contactor detects no power from hookup, it connects the sockets to the Interver; when there is power from hookup, the contactor gets enabled and the sockets are connected to mains power. This is all automatic and the switching happens so fast power does not get interrupted to the devices plugged in.
I've attached a PDF showing how this is all wired up on my van (I've actually got a number of different 240V circuits, same as you would do in a house, and one of those circuits is on this Hookup/
Inverter Contactor auto-switching system).