I had a similar experience on my '80 Chrysler. If the car sat for two weeks, the battery would be too weak to spin the starter, if used every so often, it was OK.
I got a friend to warranty the battery and thought that would cure the problem--it didn't. I started unhooking the battery after the new one went down too. One night, in a quiet shop, I hooked the battery up and heard a "chingggggggg". I did it again a few minutes later with the same result. I unplugged the chime/warning module and no sound when the battery cable touched the battery terminal. Obviously, the module had some sort of internal electrical situation as unplugging it cured the problem.
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