When you are plugged into shore power your converter is powering your DC stuff via the AC input along with the battery(s).
Sounds like, as has been mentioned, either a dirty or loose connection in the circuit from the battery through the various other connectors before it gets inside the trailer.
You don't give the age of the trailer or the battery but keep in mind a failing battery can do some weird stuff and a "fully charged" battery doesn't necessarily mean it's in top shape - just in case
Oh! And welcome to the forum!