Maybe I can add some clarity to all this (then again, maybe not)
Motors work best when there is plenty of reserve current available so they can start quickly. They also run most efficiently, i.e. cooler, when the voltage and frequency are matched to the motor design.
The motor in an RV air conditioner is designed to run at 120V and 60Hz. If the voltage is low but the frequency is still 60Hz, the motor can't turn at the speed it is supposed to, so it draws more current to try and compensate. More current means that the motor runs hotter. Heat is what destroys motors. If a motor runs continuously too hot, it will fail much sooner than it would if it were running as it was designed. That is why low voltage at a campground causes your compressor to fail. Trying to cool your RV down to 68 degrees when the temperature outside is 115 has a similar effect, it will kill the motor.
At the instant the motor in the A/C is turned on, it can require up to 6 times the current it uses when running. As soon as the motor starts turning, it starts using less current. The current drops to the rated value as soon as the motor reaches full speed, usually in about 1 second.
If a motor doesn't have enough power to start turning, it continues to draw huge amounts of current. Depending on its design, it may burn up in a matter of few seconds.
There have been several questions brought up about why manufacturers don't tell you this stuff. The rating on an A/C unit, and any other appliance for that matter, is used to size the wiring and the circuit breaker or fuse. All this other information about requiring more current at start up has already been taken into account with normal household and RV wiring and the circuit breakers. For the most part this extra information isn't normally relevent and would only confuse us.
Even in a house, when the A/C compressor kicks in, the lights dim for an instant. This is the result of the high start up current to get the motor started. The lights immediately go back to normal a second or so later. If the computer you are using crashed during the A/C start up, that is a different issue, maybe you need to upgrade the wiring in your house if that happens, or buy a UPS.
Now to the Generator:
When using and sizing a generator it is often difficult to find good information. I have talked to electricians that don't have a clue about how to size them (very scary, especially when they have been hired to install a generator transfer switch for me). There actually is a lot of information available but it is often described in mathematical terms that are not particularly easy to understand. It also gets complicated very quickly and it doesn't always translate into easy to follow guidelines.
I am going to try an analogy, I don't know if it will work.
Think of a generator like a Microwave Oven and the power coming into your house like a regular oven. With the regular oven, it takes about an hour to cook a baked potato. It also takes an hour to cook 5 or even 10 baked potatoes. The oven cooks using a constant temperature. It will reach 400 degrees and hold it there whether there is 1 potato in it or 10. This is sort of like the power in your house, ask for it and its just there.
A microwave oven will bake a potato in something like 10 minutes. Add a second potato and it takes almost 20 minutes. The microwave oven has a constant amount of power available, if it has to heat up twice as much food it will take twice as long.
A generator has a limited amount of power, just like a microwave. If I need to cook 2 baked potatoes in 10 minutes I need a 2000W microwave instead of a 1000W model. If I need 4000 Watts, even for an short time, if the generator can only deliver 2000, I am going to have a problem.
One thing that generator has that the microwave doesn't have, which is common to all rotating machinery, is inertia. Within limits the generator won't change speed immediately (if it does something will probably break) so it will actually try and supply the requested current but not for very long. The generator speed will drop quickly and the governor attempt to compensate by more throttle, but hopefully the motor has already started to spin by then. I have never seen this rating on a generator other than the vague "peak power" which provides a clue to its capability but doesn't provide many details.
A larger generator not only supplies more power continuously but has the ability to provide even more instataneous power to start motors. Not only does the engine deliver more power, the rotating components weigh more. So, for a motor that needs 1500
Watts while running, it may require a 3000 or 4000 Watt generator to start it.
For the record, inverter generators can do the same thing as conventional generators but they use capacitors to store the extra energy needed to start motors and they can compensate for the falling speed of the engine electronically. They have other capabilities that make them really desirable, but that discussion is best saved for another post.
By the way, motors aren't the only electrical devices that demand lots of power at startup. Ordinary light bulbs do the same thing. They to are almost a dead short when power is first applied and quickly go to their rated current as they heat up.
I haven't said a lot about what you should or shouldn't go out and buy.
I recommend you talk to your generator supplier, assuming it is a factory authorized dealer, they should be able to provide you with guidelines and recommendations to better select the generator for your application.
I also recommend you shop around and get other opinions.
I have seen good information in owners manuals and sales literature about sizing generators and what what types of loads they can handle. Ask to see the owners manual when at a dealer!
Unfortunately the sales people at the "Big Box Stores" and many low priced internet sites don't have a clue. The floor model may or may not have the manual with it.
Do I have a generator? Yes, it is a contractor grade unit that I bought when the lights went out a few years ago in Michigan. It powered our house in Southeast Michigan just fine but it was a bit noisey. It wouldn't run the pump at our house in Northwest Michigan so it sort of sat around. It will run our new house in Alabama, or the camper with its AC on.
I haven't decided on what I want for the Airstream and camping.