Your refrigerator will not run on electric if you disconnect the battery. The converter charges the battery but does not feed the refrigerator directly. That comes from the battery itself.
As fulltimers we've been plugged into electric power for up to three months without disconnecting and not had any battery problems. However, we have more than just the refrigerator and some parasitic items, like the detector (co2?) in the kitchen, drawing power. I don't know if that's enough to make a difference. Maybe someone else can help on this.
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