I use the fridge on propane (not in auto) while charging. (and of course all the time when not charging too since the gen is off then)
The Honda 3000 is maxed out for watts using all my chargers, so if the fridge had to be on 110v via inverter (charging takes too long to just leave the fridge off that length of time in summer) I would have to use fewer charging amps and a longer gen time. Also there is the amps draw at the same time as trying to put amps in making it all take longer. I have not quantified this to see how much longer etc. Tapering of amps later in the recharge would allow some spare 110 so it is a race if the amps taper before the ice cream melts I guess.
Solar would confuse things :) The trade offs get really complex!
Solar in winter to heat the water in the day time, that would then keep the batteries warm overnight? Thinking you get more AH saved by keeping the batteries warm than you would get from the solar supplying some AH instead of using so much of the cold batteries
Or just get AGMs and keep them inside and hope they don't ever vent.