The refrigerator doesn't "need" the genny, it will run off of propane while on the road, but if your refrigerator has an "Auto" setting, it will run off of electricity (typically 120 volts") then switch to propane when the electricity is not available.
A 5.5 kw generator will run at least two ac units and any thing else you might need with the possible exception of the electric water heater.
It is OK to run the generator while going down the road. I have done that more than a few times, such as when crossing southern Idaho in July, outside temps of 105 degrees. Usually start the genny a couple of hours before an overnight stop so the trailer is cool. When it is that hot, it can take a couple of hours to get the inside of the trailer comfortable.