If the battery or alternator is bad, the PCM is not happy. Even at 32* you should have at least 15-20 minutes of reserve power to make the starter turn over the engine non-stop. Running the starter 15 minutes non-stop will ruin the starter, but since your battery is all done after just a few minutes, it is likely toasted.
How old is the battery ? Typical battery life is 3 - 5 years.
Load Test the battery and the alternator.
Perform routine maintenance of the start and charge systems. Remove, clean, and firmly reconnect all the wires and cables to the battery, starter, and alternator. Look for corroded or damaged cables or connectors and replace as needed. Copper wires should be copper color, not black or green. Battery terminals and battery wire connectors should bright silver, not dull gray/black and corroded. Do the same for the grounding wires from the starter to engine block, the ground wires at the coil, and the ground wires from the battery and engine to the Jeep's frame/body. You must remove, wire brush, and clean until shiny the cable/wire ends and whatever they bolt onto.
Jeeps do not tolerate low voltage, bad wire connections, or poor grounds.
