There is no way on earth you will be able to run an AC on a 1000 watt generator, for although the AC running continuously uses only around 700 watts of power, every time the compressor starts, it will draw around 2,000 watts for a couple seconds, and that is beyond the capacity of a 1000 watt generator.
Add in the fact that your converter will be drawing something to recharge the battery (amount varies due to the state of charge on that battery), and even a 2000 watt generator becomes very iffy indeed.
At a bare minimum, you need 2400 watts. But that is still going to be taxing that generator hard, thus likely shortening its useful life. I would recommend strongly nothing smaller than a 3000 watt unit.