As someone with a similar setup, if I were to do it all over again, I'd pay higher price and buy a Honda eu7000i, and not go with a 3000 watt unit for a 15k, or even a 13.5k BTU A/C.
My 3000 watt generator runs my A/C unit... but I'm sure that it bogging down when the compressor starts is not doing any favors to the A/C's lifespan, and factors like the gasoline type and fuel additive can mean the difference between the compressor starting, versus it bogging down and overloading. With a larger generator that can handle the 60+ locked rotor camps coming at it, even with eco-throttle on, one can have the A/C unit run effortlessly, as well as handling the wattage at a lower RPM and noise level. Of course, a higher wattage generator allows more stuff to be run.
Once I go to a motorhome, even though it will have an Onan genset, I will definitely have a Honda eu2000i in one of the bays, just because it uses far less fuel than the Onan does when running it to charge batteries. Plus, it makes a lot less noise and can be set away from the motorhome so it doesn't vibrate it. Of course, if the MH batteries are dead, the Onan won't start, so having a generator which can start without needing a battery is always useful.