I like to store our rig indoors over winter with a full gas tank and full LP tank just in case the rig is outside but we need the rig warm inside. Or in emergency we need power for the main house. Like most of you, our rig is a 4000 watt portable power station with 40 gallons of replenishable fuel.
But there is no technical reason why LP tanks need to be full or even empty during winter.