Set light to minimal, cover the light sensor to lower light to minimum (optional), let battery-eye run and remove the plug.

As you say, it should keep running. After a while, say, an hour, run battery-eye and see what it said. See if the graph is descending like it should (could screen-shoot it and post, Ctrl+Shift+P to do a shot, it's in ~/.images/screenshots IIRC).

Then turn off the backlight. If it goes black and non-responsive, let it do that a while (maybe an hour), then boot it. Battery-eye should let you know if there is a great drain, of the battery signals low or anything like that.

Don't know if that will help, but a battery signaling "OMG" will power the device instantly and have it refuse to boot until plugged.