No... The inverter won't run the refrigerator (presuming you have a Norcold AC/Propane refrigerator). The inverter is only hooked to a limited number of outlets such as the TV outlets, the bathroom outlet, the outside outlets near the side entry door and an outlet near the hot water heater switch.
Your propane tank contains 10+ gallons when full (to 80% tank capacity which is max) and will have enough for literally weeks of use for the refrigerator. Obviously, if you use the propane for heating, for the stove or to make hot water... The time will be less.
I have made it 4 or more weeks in the summer on a full propane load.
Lots of places sell propane so not a problem getting the tank refilled in MY experience.
Worst comes to worst, run the generator, it will run the refrigerator just fine. There's about a 1mpg gasoline penalty.
OH, we always set the refrigerator to AUTO., if shore power is available (from generator or you are plugged in) it will switch to AC, otherwise, it will use propane. No need to mess with the setting.