Thanks for all the advice.
The problem was not because of a drained battery.
I finally got it working. It seems that one of my mistakes was to pair the Petrel with my computer through Windows before attempting to download the logs through Shearwater Desktop. I should have let Shearwater Desktop do more of the work. In fact it only works when the "Bluetooth Network Connection" in Windows is shown as "Not connected", and the Petrel does not appear as a device in either "Devices and Printers" or the "Bluetooth" section of "Device Manager".