If you are 100% certain it is not the transformer, or connections, that only leaves the battery.
When it clicks and goes open circuit, disconnect the charger and check the terminal voltage of both.
The bad one should be 0V, or close.
There was a recall a long time ago. The wires inside that connected the terminals to the plate, or one cell to the next, or something in there, were defective.
The disconnect/reconnect caused sparks inside, which could potentially cause an explosion.
Not something I would want to dinker around with.