Hmmmm so you are into it for $131, just about the cost of a PD-4045, but that's yesterdays story..... To answer your questions.
You normally hard-wire the converter to the battery
so that every time AC power is applied, and the converter is turned on, the battery
also gets 12 volts. In short, I wouldn't ever allow the battery
charger to be the sole source of DC power, it should always be connected to the battery. Remember, it was sold as a "Battery Charger" not a converter.
Overnight I did remember a problem we have with our Airstream when we full-timed many years ago. It had a battery charger and a monster 8D battery for the DC power. However, we could not use the 12VDC TV, the AM/FM radio or the CB Radio when connected to shore power if the battery charger was ON. Battery chargers may not have a clean output and, in our case, it put a buzz on the DC line. This might also cause some issues with things like computers and phone chargers etc. Watch for it.
Because each branch breaker protects a particular branch and the Main breaker limits the total capacity of the input cable, the added up value of the circuit breakers can exceed the main. For example, you wouldn't use the 15 amp outlet breaker for a 1500 watt electric heater at the same time as the 20 amp breaker for an A/C.
Again, because of the parts you are using, I WOULD NOT connect the DC ground to any ground common to the AC side... period... Set up a separate ground point for all your DC circuits and connect the negative side of both the battery and the battery charger only to that point as well.
Bottom line... if there's any chance you can return these parts, that would be my #1 suggestion. #2 suggestion would to start a fresh thread.... lol.