I installed a 1500 watt unit in my Ole Toyhouse years ago. Mainly to run the microwave and also make morning coffee.
We didn't have a genny, and only had a standard Group 27 Walmart marine battery, so we couldn't run the high loads to long, maybe 15 minutes or so. (We'd run the truck to recharge the coach battery)
My 1500 watt would not run the 5000 BTU AC unit. It would run the blower, but couldn't start the compressor, the inverter would kick out with an overload error.
An interesting side effect of RVs, a lot of people learn about energy conservation, and get an awakening on how much it costs to generate electricity.
Here's a link to my old Toyhouse inverter install..