FOLLOW UP:

OK, so, I let it sit untouched for a week with the new Diehard installed. Saturday, it fired right up, no worries, so I'm reasonably certain that THIS TIME it was just a dead/dying battery.

FWIW, here's what I have observed with my dash displays:

The clock stays on 7/24. The temp and odometer go out a minute or so after the key is removed, and come on as soon as I open a door.

When I first got the car, the service reminders had not been reset, and the "double wrenches" icon came up and displayed how far PAST recommended service I was when I started it (nag), returning to the standard odometer reading after thirty seconds or so.

Now that I have reset the service reminder, to see the service reminder "wrench" display (how far UNTIL next service recommended) I have to press the left-side dash display adjustment ***** - it never comes up on its own.

I'm going to assume at this point that everything is normal.
Happy me!