First of all, if there was a short in the shifter buttons I would expect errant shifting behavior. There is a third button that is intended for flight deck and if that wire is shorted it could cause a low constant draw. There are certainly resistors in series and it would just be a trickle. Was battery life normal before you had this problem?
I would be somewhat concerned about identifying the source of the problem before just replacing the battery. Although at $10-$20 a piece I guess you can afford to experiment. The battery itself should not be particularly expensive. The shimano battery is only 800mAh
I am not an expert on battery chemistry but I have programmed charging circuits before. These batteries are probably two cells that should achieve roughly 4v each or 8v total at maximum capacity or 3.75 'nominally'. There are several stages to charging a lithium battery. In the first stage, the charger supplies constant current as the voltage rises. When the voltage reaches the peak there is then a slowly decreasing current saturation charge at constant voltage. If the current fails to rise in the first stage the charge will be aborted. This is likely your problem.
I believe it indicates chemical or physical damage to the cell which reduces the chemical storage capacity. This could've been caused by one really bad short which drained the cell too quickly if the overcurrent protection circuit was somehow insufficient. You might double-check the rest of the wiring in your seat-tube. It would have to come from somewhere in there, not beyond the front wiring harness. I would also see if the battery is abnormally hot when charging as this would also terminate the charge cycle. Is there a chance that it got wet?
As far as a source of batteries goes, sparkfun has some polymer lion batteries that weigh only 3g and store more energy than the shimano ones. You'd need two and a special charger that can handle two. They are only $7 each.http://www.sparkfun.com/products/731