Well, the situation is looking a bit grim.
My understanding is that "sleep mode" not only clears the screen, but also makes it unresponsive to input. Power will still be consumed, but at a slower rate than it would if the reader was being jostled about in a backpack and "reading" signals through the screen and responding to them.
I don't normally use my reader to listen to MP3s. I explicitly put some on there to help drain the battery one time in order to help calibrate it. Unless it is spontaneously launching MP3 files without my deliberate intention and without my knowledge, this doesn't seem to be a likely explanation. Less so, give what I just discovered.
And I'm aware that USB ports can power down during hibernation cycles - but that did not occur for the four hours I just devoted to charging the unit via USB.
And once I disconnected the unit from the USB it first booted up as if to start, but the immediately gave me the "No Battery!" warning it did previously (yes--I took note of what it said this time)!!!
All subsequent attempts to restart the unit gave the same warning/message, and it just shut down.
I decided to try a "hard reset" - powering up with the Previous Page (<) button and Volume+ buttons pressed simultaneously, to see if it might be a software glitch. Fortunately, most of my books were loaded on a 4 GB SDHC Sandisk "Ultra" (15MB/s) card. I had to connect the unit to an AC adapter to do the reset, which I've done. After the brief period it was connected to the PSP-style charger with the unit reset, and following the screen-tap calibration and language selection routines, I disconnected the adapter.
The bars fell immediately from 4 to 2. I wanted to see how quickly I would get a low batter or "no battery" warning, so I started doodling in the Handwriting app, and within a few seconds it dropped to 1 bar. I saved that drawing, and started typing this post. The meter still held at 1 bar that whole time. Then, I tried a second "doodle" and watched the bars "flicker" as if the screen was refreshing as the unit does during page turns to clear ghosting, and the bars suddenly jumped back up to 4! At least...until I clicked "Done" on the second doodle, at which point the bars have gone back to 1.
This is some very strange activity going on here. With the unit at 1 bar I put the SD card back in and it initially flashed a "No Battery!" message in the status bar at the bottom of the Main Menu screen, followed by "loading" that the "yin/yang arrows" graphic that comes up during the start-up and re-pagination sequences. I'm able to open some of the books on the card with the battery meter holding at 1 bar.
So, I guess I'm going to try using the reader off the PSP charger for awhile, just so I can get a little pleasure using it, during which time it should (theoretically) charge.
My own experience with both this reader (prior to today, at least) and the previous PRS-600 that was replaced due to a fading screen, as well as reports I've read on MR, all suggest that charging from a properly-rated PSP style charger works as well, if not better than off USB. I believe the Sony rep I spoke with earlier today meant well, but she was probably working off a decision-tree script and may not have any real world experience with the device. I welcome feedback that demonstrates I am mistaken in this assumption.
Heck--I welcome any feedback that might lead to a solution here!
@dwanthny: Thanks for your suggestions, even though I don't believe they were applicable in my case.