If you get really small appliances the 1000 might work. There are trailer sized tea kettles, coffee makers and if you get a 2 burner hotplate the smaller one would work. It's all in the wattage for resistance type devices. I ran a toaster oven on the 1000 by using a variac but that gets a bit complex although it worked very well with a form of porportional control.
Inductive devices (Motors) vary depending on the inrush which was killing the 2000 for the fridge. Once the fridge was running on the 2500 watt genny it only drew 3.5 amps, even the 1000 would have handled it but not the compression load.
Freezers draw even less IIRC. Put the bodies in them.
Stick an amprobe on the 2000 and try your fridge. Mine is 25 years old and yours may differ. It would be good to know.
The next size up is 3000 watts and it starts getting heavy.
there in actually li ays the answer, read the wattage draw on the appliance and the startup draw if labeled or fine the spec someplace. My brothers cottage on an island has had the 850w 2 stroke gen keep the fridge going all weekend, 4 yr old fridge. A blow dryer takes more wattage than his fridge. The 600w microwave will run , but not while the fridge is on, and even thought its an 850w gen, it actually only puts out about 725 continuous.4
Please people, if you want to invent your own grid and switchouts, do it on an Island where your the only building on your grid. Not having an actual generator disconnect switch and maybe even an electrician over to check stuff out is just a really bad idea. Thinking the breakers will protect you will get you or worse somebody else killed.