Unable to get Generator to power Air Conditioner

You are underestimating the startup current demand on the Dometic 13.5. It can be in excess of 40A, but they are not identical and even with the same unit the current requirement varies with temperature and with how long the compressor has been off. This isn't the first report that a 3KW generator failed to be able to start one. 

A typical residential circuit breaker is designed to hold at least 7x its rated current for a few tenths of a second, which is plenty long enough. Depending on its design and protection electronics a generator might or might not be able to do the same thing. Generators store energy in their rotating mass to handle surges, but inverter generators have to be able to pass that power through their electronics without damage. Briggs is limiting those currents more that Honda is, proportional to the continuous power ratings of its generators. That is not a coincidence, I would bet Honda uses relatively higher capacity components than Briggs does. That would increase their costs but provide a higher level of reliability, and that's what we see: Hondas cost more but have a top notch reliability record.  

The Easystart reduces that 40-ish A startup surge by about 2.5-3X, well within the range of a Honda 2K. 

If you're close the Dometic starter might get you over the top. But of course it might not, sometimes those starters are designed to help the compressor motor get started at the expense of even higher line load. Good luck and let us know how it turns out.

Many years ago I bought the smallest Generac inverter generator, rated at 800 watts IIRC.

It would barely run a 100 watt light bulb.

Junk it was.

Question in line with this thread topic. If you have a 3k generator AND you get the AC cranking how much capacity is left on the generator to run other appliances, lights, etc, and will your onboard battery be able to charge? I am looking at a gen set to charge my batteries primarily, but if the temps rise or we are camping without tree cover the AC would be great to run in the evening or during my kiddos nap time.