I'm genuinely puzzled by this and indeed similar stories recounted at other times.
I cannot see how a battery could fail within ten minutes of takeoff simply because the alternator fails. There must have been enough charge in the battery to start the engine and even a small amount of remaining capacity would be enough supply radios etc for more than 10mins.
For instance even as little as 10% capacity (unlikely for an aircraft in regular use) would still be about 7AH (Ampere hours) which would require a current flow of 42A to completely discharge in 10mins. For the undercarriage retraction to explain that would require very high currents to flow.
Even if some load had discharged the battery, switching off all the avionics, lights and so would allow some recovery in battery voltage, surely enough to work the stall warner which is just a buzzer - 1A or so and probably works down to 6v anyway.
So I'm convinced there must be some other factor in play - a battery fault or an undisclosed load that can handle 50A or so without making a burning smell. Fan belt failure just doesn't meet the facts as described.