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I think it most likely that the charge controller chip that kobo likely employs is kaput or that there is some other problem in that circuit. It could be as simple as a bad trace, bad solder joint, etc. If you can't get Kobo to fix it, you might try a local HDTV or computer/tablet/cell phone repair shop, or try ascertaining which chip is a Li-ion charge controller by looking up the parts online. Now, if Kobo has the microcontroller doing the charging then you may have a firmware issue, but I think it rather unlikely... Alternatives include external charging of the battery, but that would doubtless be painful. OTOH, a small connector could possibly be added to the case for just that purpose and charging pereformed while connected to the reader as long as the reader doesn't interfere with the charger.
I'm pretty sure you'd need at least a three wire connection to the battery to have a data line that could be used to interrogate a battery chip to ensure a certain battery was employed. At least I can't think of a way to do it without getting highly overly elaborate, if it could be done at all unless you're going to employ a radio or the like for just that purpose. Too silly. By highly overly elaborate I meant super-imposing an AC waveform on the DC power leads or somesuch nonsense, or having an active switch(MOSFET?) to temporarily internally disconnect the power lead from the actual battery so it could be used as a data line, etc. Then you'd need a good sized cap or super cap to temporarily power the reader... How would such a data gathering process be triggered? blah, blah, blah highly overly elaborate. Two wire connection likely equals dumb battery.
Good luck.
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