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I'd suspect the converter box before the battery, unless the battery is drawing a LOT of current, or you've got significant dc loads (much less likely), or the battery clamps or connections aren't tight.
How did you check the converter? Are you plugged into shore power when the lights dim?
Have you checked the clamps on the battery to make sure they're tight? If you have a voltmeter, check the voltage across the battery terminals, and then across the clamps. If they aren't the same, tighten the clamps.
Check the ground from the battery to the frame, too. I found that loose when I first got the Montana.
Bob
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