Yes, it turns out the screeching sound was definitely the gas leak detector. It makes that noise when there is either a loss of power, or when power is restored. Accompanied by a yellow warning LED. Several months ago, we would get the sound whenever we pressed the "store" button while getting gas, although it didn't make the sound when switching to "use", until this recent incident. It's mounted under the bed on the driver's side, about 12 inches from the inverter/charger. Unless you've got your ears right near the detector, down near the floor, it still sounds like it's coming from the inverter/charger. Strange acoustics in the coach, I guess. But I will be changing this out before we head out on the road again.
Thanks for the info about the Safe-T-Alert, George. I forgot to get the CCI model number today, and there are several variations of the 7770, with variations in the Safe-T-Alert replacements. I'll be ordering one when I know which model to get.
We ended up replacing the coach batteries. Possibly due to my neglect or inexperience, but definitely beyond hope; but I'm not ruling out a bad battery (at least one very bad cell).
Ron Dittmer: fiddling with the plug behind the wall-mounted Tripp-Lite remote was more than the wife wanted to deal with, so with the new batteries, we had a cut-out installed, purchased from Napa Auto Parts; it's easily accessible, and meets the requirements of the family-mandated KISS principles. Because of the battery compartment configuration, it had to be installed on the positive terminals, and they had to account for the fact that the batteries are wired in parallel. But initial testing indicates that it is working like we want it to. If anyone is interested, ask me again in 6-months if we're still pleased with this set-up!

One other point about my inverter/charger: it doesn't have a plain ol' on/off switch. It has a 3-position slide switch, Auto-Remote, DC Off, or Charge Only, in that order left-to-right. It's set for Auto-Remote, been working fine for the past 8-months, so will leave it be.
I really appreciate the pointers to the gas leak detector. Since the manual didn't mention any audible warning sound, I didn't really look at that as the source of the problem. Greatly relieved that it's nothing to do with the Tripp-Lite!
Thanks again for all the input, feedback and hand-holding. It's nice to know we're not alone out here.
Cheers and regards,
--GeorgeB
p.s.-- so now I count 4 George's. Can't decide if that's good or bad...

UPDATE re: master "OFF" switch on the Tripp-Lite: something kept nagging me, so I went back thru the owner's manual I'd downloaded for the PowerVerter MRV DC-to-AC Inverter Charger; I'm guessing/hoping that it is the same as what I have, crawling around on the floor trying to get the m/n was too tough, expecially considering eyesight & lighting. Anyway, that manual refers to the "Operating Mode Switch" which I'm pretty certain is what Ron Dittmer was referring to, and has the following regarding the "OFF" position: "OFF (most models)", and "DC OFF (select models)", obviously I don't have a "most model", but a "select model" -- whoo hoo!! Now we know why I was confused. But I'm still not changing any settings. For the foreseeable future, anyway.