Go with a 3000 watt generator, otherwise you are barking up an expensive tree.
High wattage loads like A/C while can be done via large battery banks for short periods of time, it draws the batteries down at a much faster rate than small low amperage loads. Basically depleting the charge at a much faster rate. It is also hard on the batteries and you need to recharge the batteries very quickly if you hope to have long life from them.
As noted, 13.5K A/C can draw 13A at 120V which is 1560W, but that does not take into account the startup surge which can be as much as 25A at 120V or about 3,000W.. Most cheapo inverters do not have enough surge capability in time they are able to provide the surge power to handle A/C compressor startup..
That 25A at 120V roughly translates to 250A or more at 12V..
Generator is the correct tool for the job.