Hi,
Getting to the specifics of running the generator with the inverter, I have made a point of letting the air conditioner start on the inverter, and then starting the generator. There is a time delay before the inverter/charger transfer switch moved the load to the generator. I don't know if that delay is adjustable or not, but I was pleased to be able to start the generator "unloaded".
When the air conditioner started with the eco switch in the on position, even though I have the "boost" model of the Yamaha 3000, voltage would sag to as low as 92 volts. I don't remember what the sag is with the generator cum Magnum inverter, but I'll probably make note of it tomorrow.
Tomorrow promises to be a brutal day with the temperature reaching 31 c (88 f) at the destination. I will be boondocking sans shore power.