Gensets can be interesting. I have an Onan 2800 watt microlite which runs an 13k BTU A/C even at altitude along with accessories. But then I have had a Onan 3000i fault on the same A/C but only when used with 50' of ten gauge extension cord.
The problems appear to be the initial compressor surge which can be several times the steady load, the power factor, and altitude.
The double Honda 2000i setup has worked well for keeping the A/C going on hot afternoons. The 3000i is iffy. Reports I have heard about the Yamaha 2400 are mixed. A Boliy 3k seems to have done well on this test at Unionville recently.
What I see is that 3kw genset capability is marginal and 4kw is needed for reliable service. The mechanical gensets like the Onans have mechanical mass to help with the initial surge while the inverter types may or may not have needed surge capabilities.
Batteries need a proper vented battery box, even AGM's (although many sidestep code with these). Repositioning them runs into many interesting problems. As you can see from Ricky's links, a lot of creative energy has gone in to adding gensets to trailers. (I've got an Overlander with a genset compartment mod but fuel supply and battery leads were obvious problems to the installer).
re: "As the A/C works, it continually cools the inside ambient air and the temps continue to drop until the thermostat is satisfied.
" -- but they only have so much capability and when heat gain through walls and windows matches A/C capability, the temperature will drop no further. If your RV A/C can hold 20 degrees colder inside than outside on a hot day, it is doing pretty good relative to the experience of most of those of my acquaintance.