So, it's been a while since I reported the final ( at least so far ) solution. Looks like both of my chassis batteries were toast. They are rated at 1050 cold cranking amps. Had them both bench tested, and they tested at 660 and 700 CCA at 12.7 volts. Don't know how old they were. No markings on the batteries. Big Boy seems fine. The button switch on the BCC switches the relay just fine.
I replaced them both, and have been monitoring the voltage for the last week or so. Voltage holding at 12.7 volts. And no problems cranking either the engine, or the generator. So far so good. Everything seems to be working.
Does anyone here think that I have only cured the symptoms, not the disease?