I may be all wet on this one, but I'll toss in my thoughts, none the less.
Is it possible that you reversed the polarity when you hooked up the battery? Since the lights work on the 12V D.C. service (even when connected to 120V A.C.), the lights would work regardless of polarity when running off the battery. But when you plug into A.C., now your battery is being charged by the converter. If polarity is reversed, the converter wouldn't recharge the battery, and the lights would dim. The 120V A.C. items - like your outlets, television, microwave - would work fine. I'm certainly no electrical authority, so I don't know if that makes sense to anyone but me. But it might be worth checking to make sure your black is connected to the + battery terminal, and the white is connected to the - terminal (like in your House, not like in your car/truck).
My apologies to the electronic gurus if I'm all wet here. I do know that electricity & "all wet" are NOT a good combination!