As Ned says, that is an exercise in futility. You lose 5-10% of battery power as the inverter produces 120v from 12v. Then you lose more as the converter converts it back. The longer you run that way, the quicker the batteries discharge because the converter draws more and more power to attempt to replace what has been lost, but instead it just makes the problem worse.
Visualize pumping water out of a tank with a badly leaking pipe and into another tank. Then use another pump, also with a leak, to pump the same water back in. The longer you run both pumps, the more water you lose. You can't gain anything - you can only lose more and more. Plus, you have to supply power to the pumps.