Honda - a 2000 will NOT be able to run your A/C even with a soft start unit - it isn't just the ability to start it, it is the ability to not overheat the generator since most RV A/C units take more tha 13.3A to run - which is the max RATED for the 2000 Honda. Now the newer 2200 most likely will work - you need to measure the AMPs of your unit while it is running - not just initially, but after say 30 min. to make sure it is at or less than 15A (rated load).
Honda - the ONLY unit that rates their "max" run for up to 30 minutes. ALL others are only allowed to be for a few seconds above the "rated" output. If you find otherwise, please let me know. I even called Yamaha and they confirmed for their inverter units.
Champion - Find a service manual for it - if you find it, please share with me - I've never found one. I called the company and they don't exist. I've stopped at "authorized repair" shops and they don't have any. The ones I'm familiar with are noisier than the Honda. A friend bought 2 of the inverter style and they were blowing black smoke out of the box (too rich).
Honda - easy to find parts AND a true service manual. Need more advice/help - check out:
https://groups.yahoo.com/neo/groups/Honda_EU2000_Generators/info
Of course some of it is your typical INTERNET help - useless, but there is also a guy "Robert" who is a HONDA EMPLOYEE who answers questions there.
I had 2 Honda EU2000 gensets, made my own parallel box and used them dry camping. The box will measure the output of each generator so I would not overload either of them. When the larger cheap house generator went BANG I sold 1 of the EU2000 and bought a EU6500 to take camping or for the house. I love both!!!
Remember, just because a generator is powering a load doesn't mean that it will run that load long term.
Also, please allow your generator a cool down period - just like your diesel - after it has had a load on it or you can melt your windings (I know folks who have done that).
There are many other reasons to go Honda and stay away from the others. I'll be glad to answer questions. I did HOURS of research before I bought my first Honda.