Look at the specs for amp draw or buy or borrow a clamp on amp meter and check the starting/running/cycling amperage of the ac unit.
A 2000 Honda will most likely run it but I would want to know what range of its available amperage I'm operating in.
2000 is the surge wattage of the Honda but the running wattage is 1600. So....1600 watts divided by 120 volts equals 13.33 amps.
So if it's powering the ac unit, and say it's pulling 7 amps which is about half of its capacity or is it at 12 amps which is most of its capacity. The way Honda words it in their manual is that if you are borderline of tripping the overload indicator you will have reduced service life.
16.6 amps is available for the 2000 on a surge basis. So again, if you are running along at 12 amps with cycling spikes 16 amps, you are maxed out.
I'm guessing you will run at about 8 or 9 amps with a start draw of about 12 or 13. If you know that you are cruising along at say 8 amps and you know that it's warm enough that the compressor is not going to be cycling on and off you may also be able to to run other small loads with it.
If you get an idea of the amp draw of your converter while charging your battery, you MAY be able to charge you battery at the same time. Again, go by the numbers as your converter will pull more amps if your battery is very low vs a low amp draw if you are just topping your battery off.