All your experiences with the 5.5 Kw Generator make sense. I had the same size Generator, but, my 3/4 H.P. Well Pump, that is over 250 feet deep, would have problems with that size generator, and nothing else on.
Everytime the 3/4 H.P. Pump started, I could hear the motor on the 5.5 Kw Generator, grind down, and almost shut down.
The starting torque for that Well Pump required a much larger Generator, that is why I went with a 15 Kw, and don’t even hear the motor load when running the Water Well Pump.
I was on City Water. They have big gensets for the pumps and priority getting power restored after the hurricane.
For this house I would prefer 7.5 kW as an economical backup. That still wouldn’t provide whole house A/C at Houston. But getting gasoline after a hurricane is not easy and big generators are thirsty.
Is your 3/4 pump 120 Volt? A 220 Volt pump should have been okay on 5 kW Gen. Of course I don’t know what else you were running.