With my trailer, I don't need the battery to run the 12v stuff if I'm plugged into shore power.. I can run all appliances and turn on all the lights and use all the plugs, AC, Micro.
When you say you hear your converter turn on, is it the cooling fan you are hearing?
That's what mine does, whether I have the battery attached or not. If I turn on a couple of 12v lights, the converter fan will turn on eventually. If I turn all of them on, the fan will turn on immediately.
If that's the case nothing to worry about.. That's what it's supposed to do. I'd be more concerned if the fan didn't turn on..
Anyway, good luck!
Mitch