I don't have enough experience running a 10k versus say 5k generator and measuring fuel burn to say for sure, but I can't imagine if not under load it matters much. They control their own throttle basically and idle if the load is low - I'd expect 10 to 20 gallons a day running a gasoline one (size/load depending) I guess at current fuel prices that's steep but not if you got no power.
By you saying you switch half the panel obviously cutting costs is a main goal, but I don't think most care at that point. An old place I had used to lose power about once a month, I'd basically shut off the electric water heater, oven, microwave until I needed it (because they would over tax the generator if everything else was running - had to plan when you ran those items) but otherwise I'd just just fire up the generator and live. If it was summer ACs were running, winter the Christmas lights stayed on. For most of us bigger is better.