There's no on/off switch? What a huge oversight in an RV fridge! If it were me, I'd wire in an on/off switch on the positive power lead, with the switch right next to the fridge.
Though, now that I'm thinking about it, are you sure you can't turn the compressor off? In mine (a garbage Furrion), the compressor can be turned off when the dial is spun all the way left, but the light still turns on when the door is opened. As long as the door remains closed, the light doesn't turn on.
As long as your trailer's charger is working properly, it won't overcharge the batteries when plugged into shore power.