The 5000BTU air conditioner should run just fine from a 2KW Honda type generator... This is not the bigger problem however as most public camp grounds have generator run time restrictions in place especially here on the East side of the US and you will only have certain windows to run your generator in. Definitely not enough time to cool down a trailer. Most times are from 8AM-10AM in the mornings and then again from 4:30PM to 7:30PM in the early evening. The NATl FOREST area may have 8AM to 8PM available to run the generator but this is not always the case. We have not found any public place that allows the generartora after 8PM at night. Let alone to be able to run them all night...
What most folks do in these cases is to beefup the trailers with more battery capacity to run all the 120VAC items you want to run from an installed INVERTER conencted to the battry banks and all of the 12VDC items direct connected to the battery banks.
Then the trick is to be able to re-charge your battery banks within the generator run time restriction times to get the batteries back up to their 90% charge state. Failure to always get back to the 90% charge state will most likely result in your batteries starting to lose their performance specs operations. You can do two weeks of this type of charging between 50% and 90% charge states and not do harm to your deep cycle batteries using smart-mode technology charging techniques... After that however you have to charge up to a full 100% charge state which will take around 12-13 hours to fully charge your batteries. This of course cannot be do at most public camp grounds with their generator run time restrictions in place. This is usually when we come home off our trips and get our batteries back up to 100% charge state at home...
It works good for us doing this type camping off the power grid but does take some initial planning and battery system beefing up.
Roy Ken