It should bog a little when a heavy load is first applied, but once the A/C is running, it should stay stable. If it is continuing to bog afterwards, most likely a dirty carburetor. It's starving for fuel under load. You might get lucky, and find that it's just a dirty fuel filter.
While this is a very popular generator, no rebuild kit is available that I could ever find. Ultimately, I replaced the carb on mine. Not overly complicated. Runs perfectly once again.
Here's a video of the job:
https://youtu.be/DUEthYuKGMkSource of parts:
Onan carburetor:
https://amzn.to/2HgqRFkOnan air filter:
https://amzn.to/2tO8C22Onan fuel filter:
https://amzn.to/2SM1KMKOnan fuel pump:
https://amzn.to/2tNtvKMOnan oil filter:
https://amzn.to/2XI4luC