I'm happy that you solved your problem so easily (it might not have felt "easy" considering all of the testing and diagnostics time but the final solution was, you have to admit, quite basic). I have the same symptoms and many of the same diagnostic characteristics but my charger is good, and I've tested the laptop with more than one charger only to get the same results.
The computer acknowledges that I have a battery but it simply won't charge it. I took the battery out, charged it with an independent battery charger, and replaced it into the tablet side of the computer (Click W35Dt-A3300). The computer shows the fully charged battery, now says that it's charging, but 1, doesn't charge, and 2, shuts down at the slightest disturbance in the AC source.
In the Device Manager, there is an option to disable the battery for to allow for service or maintenance (maybe it's in the BIOS menu?) and I wonder if this has happened and now needs to be reenabled? I don't know how to enable a battery since there doesn't seem to be the opposite to the Disable option.
Have you any more ideas? You obviously have the diagnostician's mind required for this type of problem solving.