Everything Adam says is good advice.
However I do think having everything 240 a better approach for various reasons getting polarity wrong being one
..but what I would be doing is get another battery to have as a spare...or a small genny.
I would hard wire the inverter to the battery...now one should not get polarity wrong ever but it has happened to me too often..late at night and tired you can stuff up...well anytime really. I would leave my battery in my car ( this is when I am coming up from Sydney) with the inverter hard wired and then run a 240;vlt lead to the gear..besides eliminating mistakes I just found it easier to set up..At that point I did not have a spare battery or genny as it was near the house so I could charge if needed.
Alex