What size and condition is your positive battery cable? You're dropping over 1/2 a volt across that cable with just the lights on - that sounds excissive to me.

A lot of the old 6 volt systms got 12V replacement cables over the years and they just won't carry the load. Some of the 6 volt guys around here use #1 or #0 welding cable with soldered terminals, or 2/0 battery cable.