We brought home the RV last night, and despite the electrical mess, it was a great trip. Hubbie started working on determining what the issue was this morning, and I was able to recreate the issue (huge load on the batteries) just by turning on three LED lights. Note that the battery level indicator lights did alert me to the fact that there was something wrong, so although it's called an "idiot" light, this idiot is thankful it was working.
We tested the (new) batteries and they were fully charged, so that wasn't the issue. The Air Conditioner worked flawlessly on shore power.
We tested the voltage in all circuits - AC and DC - and both using just shore power and again with just the batteries. All a-ok. The problem didn't show up until we applied a load to the system.
We were told by a fellow camper on Day 1 that he has seen these issues when the ground wire was corroded (and 'thank you' to "wa8yxm" who also mentioned a ground wire connection). We looked at that first and voila, our ground wire looked corroded. We changed out some connections and tried to recreate the problem and couldn't. Electrical fixed.
So in case someone runs across these issues in the future (large unexplained elec load), look to your ground!