Your 2000 Watt generator should be able to run your microwave or your AC without issues. Only one device at a time though. Do you know what size AC unit you have?
What is probably happening is that your battery is dead so your converter is drawing power to charge your battery and your fridge may be switching over to AC when the generator is starting up. Add your AC or microwave on top of those loads and you will run out of power.
Our 2000 watt generator will run our 13,500btu AC unit but everything else has to be off in the trailer. Switch off your converter at the breaker and make sure your fridge is off or on propane and test your generator again. And make sure your generator is not in eco mode if it has that mode. If your AC is the same size it may or may not be able to run the AC. Ours is borderline and we did add a hard start capacitor similar to this...
https://www.amazon.com/SPP6-Capacito.../dp/B0002YTLFE
As for the battery you should definitely get more than 3 hours use out of it. I would bet you are never getting a full charge on it and if it has been run down multiple times it probably needs replacing.
Do you disconnect it while in storage?
Are you ever plugged in to shore power? Even plugged in your battery will probably take 12-24 hours to fully charge.