My camper has a built in Generac 3600 LP generator, and I also carry a Yamaha EF1000iS. The Yamaha is used about 95% of the time, but I do have to manage the power loads when I’m using it.
Running the fridge in AC or DC mode, and the battery charger when it’s in full 45 amp mode at the same time is more than any 1000 watt generator can handle. Once the charger amps are down to the point that the generator throttles back, then I can put the fridge back on AC if I want.
My Yamaha 1000, and the comparable Honda 1000 have a running wattage rating of 900 watts, and I’ve verified mine will do it many times. I’d bet your 1000 watt generator running watts rating is more like 800.
You will probably be happier with a 2000-2200 watt unit if you are going to be regularly lifting it in and out of wherever you store it. A 3KW unit is usually well over 100 lbs, vs about 50 lbs for a 2KW. For truck camper use, Unless you never have to lift the 3KW, I would rather parallel two smaller units, than deal with one bigger unit. That’s just asking for back troubles, for me.
:):)