That big piece with the blue circle is your automatic output relay. When the relay gets power there is a part that moves to make two pieces contact each other. In your picture that moving part is Y shaped piece right on top. When power is applied the arm will lift up and close the two contact points (circled in the picture below).

Click For Full-Size Image.
If the relay was working correctly the bottom left 14.28 volts would travel straight to the right through the moveable contact points and apply 14.28v to that 12.66 volt wire and charge your battery.
Based on your voltage readings you are plugged in or the generator is running because you have 14.28v on some of the wires. That tells you that the battery charger is working. But the relay is not closed and therefore no charge power is being provided to the battery.
Keeping your fingers away have somebody plug and unplug the RV's power cord while you watch the relay. Focus on the circled part of the relay, that is your contact points. You should hear a click and see the relay open and close the contact points.
If there is a click and you see the points close but no 14.28v on the bottom right. Disconnect the power, and clean out anything between the contacts or anything keeping the arm from moving. If the contact points are pitted, you can try a small file or some sandpaper to flatten the contacts so they make a good solid connection when touching.
If no click then I think you have a bad diode and are going to wind up having to replace the whole unit. In order to work the relay needs a ground to energize the coil and move the arm. My thinking is that the 2.61V wire is supposed to go to ground and activate the relay. But there a bad diode (basically a one way switch) allowing voltage to leak back up the ground and keep the relay open.
Inspect the little circuit board that the 2.61v wire is connected to. Look at both sides. Are there any dark brown or burnt spots on the board? If possible can you post a close up picture of both sides of the board?
Actually thinking further I wonder if the contacts are not closed because the battery is fully charged. So with that thought with the trailer still plugged in turn on the furnace so the blower starts or turn on a bunch of 12v things. The idea is to get the 12.66 v to start to drop. At that point the relay should click and apply 14.28v to the charge wire.