My 2KW Honda EU2000i will NOT run mine sitting on top of my POPUP... I have the 13,500 BTU model.
I never could find any public place that would even allow me to run my generator more than a few hours a day here on the east side of the US. With this in mind I beefed up the battery system to run all the 120VAC and 12VDC toys we wanted to run in a one day/night camping run off the power grid. I only use my 2KW generator to re-charge my batteries when I am allowed to run one. With the short run times allowed I can only make some coffee or a quick cookup using an electric skillet maybe... Everything gets operated from Inverters and direct connected to the battery bank.
Most public camp grounds allow two hours in the morning and maybe three hours in the early evening. The Natl Forest sites are starting to allow all day from 8AM to 8PM but only allow certain type of forest approved generators... No where can you can run a generator all night long.
Public places must be different around your locations...
Roy Ken