1000 will do fine until the battery gets low. 1000 is good for 30 to 40 amps into the battery and cut that by about 1/3rd if running the 350 watt fridge. If your converter is a typical 55 amps it will overload your generator with a low battery. BTW that 1000 is probably the surge good for 10 to 30 minutes then 900.
So keep the fridge set to propane and see where that goes. If you like the small 1000 you can get a smaller converter. Otherwise you are looking for 2000.
If you want to go deeper we can talk about power factor and the difference between wattage and the actual rating in VA.