As a followup to Little/no charging but alternator and battery test good, the battery is only charging when the charger has a significant load in parallel with it (e.g. turning the key to the on position). When charged this way, the battery reaches around 13V and works for several days of starts before dropping too low to start the vehicle. The alternator does not succeed in charging it, and without a load, the dedicated charger's control logic immediately switches to "float" mode (and the battery immediately reaches 12.5V, but drops to 10V as soon as any load is applied).
What kind of state could the battery be in that causes this? And is it recoverable (e.g. with a suitable charging procedure) or does it just need to be replaced?