The 4000 watt generator would be more than sufficient to run the air conditioner...and probably everything else in the RV. It would be essentially like connecting to shore power (assuming this is a 30A RV, which sounds very probable).
If this is a contractor-style generator, it will be pretty loud, and using it for long periods of time would not go too far towards making friends at campgrounds. Expectations may be different at other locations or if you're boondocking on public land away from any other people.
For truck stops and similar businesses (and many campgrounds, for that matter), the most accurate way to find out if they have room is to walk in and ask the clerk. At Wal-Mart stores and similar businesses that permit overnight parking, asking a manager or at the customer service desk or similar is the best rule.
