Yes the battery is like a limiter to the charger. It provides a constant voltage supply where as the charger may max out if you have a lot of things on at one time. A battery can supply much more current when there is a large load, everything on at once, than the charger can. Also the charger will put out as much as 14.7 volts. if you were to turn on just 1 light under this condition the life expectancy of that light will not be long.
As for the trailer brakes they work off the TV electrical supply under normal driving. However if your trailer was to come loose of the TV while traveling the trailer battery will then put the trailer brakes on at FULL POWER once the cable attached to the TV has pulled the pin out of the emergency switch. These 2 12 volt
systems have nothing else in common.