There are only 2 wires that connect the Univolt to the 12 volt
system in your coach. The + and -. The Univolt is connected to the + and - buss on the fuse panel. There is NO. Switching device in the Univolt!
The battery connects to the same + and - buss. As does the charge line from the TV. The + and - buss is connected to each of the fuse holders one the "line" side. Each branch circuit is protected by it's own fuse. Some coaches have a large fuse (30 amp) in both the positive and negative leads of the battery.
The Univolt doesn't know the difference between your battery and any other 12 volt
device in your coach. It does not differentiate. It is only a power source rated at about 30 amps on the output.
Since the 12 volt
devices seem to work when you have battery power the aforementioned fuses are probably good.
The internal resistance of the battery is what limits the charge current. There is no kind of intelligence in a Univolt. Thus the reason these units tend to cook batteries.
Since you are a EE the use of a multi meter should be second nature to you. Trouble shooting this system only requires a basic understanding of the system.
What voltage output do you get from the Univolt when it is powered?
What voltage are you getting when the Univolt is not powered.
If you disconnect the battery from the system and the Univolt will not power the lights etc in your coach there is a problem.
You could use a battery charger to confirm this. Leave the Univolt unplugged, connect the battery charge to the + and - terminals on the fuse panel and see what happens. It won't be necessary to have the battery in the circuit. The charger should light up everything in your coach.
I think you are reading too much into what the owners manual states. In short it says the Univolt charges the battery. Don't leave it plugged in when you are not using the coach because it WILL cook the battery.
Here is a basic coach wiring schematic.