Ukhants, CONGRATULATIONS on your Precept. Have a ball.
I'll let a more knowledgeable source answer your question about the inverter. My less than educated guess is that NO it will not power the fridge.
As for using LP to power the fridge the manuals will recommend against it....there is a liability issue. Like Stoneman mentioned the consequences of having the LP fueling an appliance can have a disastrous effect.
We have used the LP for the fridge while on the road. Prior to pulling in for fuel, we shut off the fridge and shut off the propane at the tank. Also, in most cases, it is illegal to drive into a tunnel with the flame lighted on any appliance.
Enjoy the ride, savor the memories,
Darrell