Does generator brand make a difference?

I bought an inexpensive generator to use with our fifth wheel which is a 2014 Solitude. The generator is a Predator with 4000 watts maximum, 3200 running watts. The generator runs one air condition (15k) for a few minutes then the breaker trips on the unit. The power cord is plugged into a 120/220v, 13A plug. The plug is a four prong twist lock. The only other plugs on the generator are two standard plugs that can handle a standard extension cord. With the above information I should be able to run one air conditioner shouldn't I? When the a/c was running you could tell that the generator was working due to a change in the motor noise. I have the Progressive electrical management system, it never stopped power to the with wheel.

Eventually I will be buying two Honda 2000's for the fifth wheel but after buying the inexpensive 4000 watt unit I question that the 2 Hondas can power an air conditioner. Does brand really make a difference in performance? Can two 2000's run an air conditioner, if not what do I need?