I think its time to take it to a Keystone Dealership and have them check it out especially since you are using the rewired factory location and a Furrion....I think your right about it not being wired to power. Try running an experimental power wire from your trailer taillamp to the camera? At least power it up from the trailer, than you will know for sure...
Maybe first see if the Stealership will fix it under warranty since its suppose to be plug & play
Keep us in the know-lol!